A  CALENDAR  OF 
LEADING  EXPERIMENTS 


Bacon  long  ago  listed  in  his  quaint  way  the  things 
which  seemed  to  him  most  needful  for  the  advancement 
of  learning,  and  among  other  things  he  mentioned  A 
Calendar  of  Leading  Experiments  for  the  better  Inter- 
pretation of  Nature. 


A  CALENDAR  OF 
LEADING  EXPERIMENTS 


BY 

WM.   S.   FRANKLIN   AND    BARRY    MACNUTT 


SOUTH   BETHLEHEM,  PA. 

FRANKLIN,  MACNUTT  AND  CHARLES 

PUBLISHERS  OF  EDUCATIONAL  BOOKS 

1918 

All  rights  reserved 


7 


Copyright,  1918 
By  FRANKLIN  AND  MACNUTT 


The  authors -are -teachers,  and  thty  consider 
teaching  to' be 'the  'greale'st  of  fun,  but  they  never 
yet  have  been  helped  in  their  work  by  anything 
they  have  ever  read  concerning  their  profession. 


PRESS'OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  HA. 


PREFACE. 


(a)  "A  boatman  sits  on  a  seat,  braces  his  feet  against  a  cleat 
and  pulls  on  an  oar.  What  forces  act  on  the  boatman's  body?" 
The  earth  pulls  on  the  boatman,  the  seat  pushes  on  the  boatman, 
the  cleat  pushes  on  the  boatman  and  the  oar  pulls  on.  the  boat- 
man. This  is  all  very  simple  to  one  who  has  acquired  the 
habit  of  analytical  thinking,  but  a  large  group  of  sophomore 
engineering  students  got  an  average  of  45  per  cent  in  their 
answers  to  the  question  after  two  weeks  of  insistent  coaching  on 
the  fundamental  notion  of  force  action,  and  nearly  every  human 
aspect  of  boating  was  represented  in  the  answers,  including 
even  the  chance  of  a  ducking,  for  several  of  the  young  men  would 
have  it  that  the  water  pushes  on  the  boatman's  body. 

(6)  Ask  a  student  of  elementary  mechanics  how  a  body  (a 
particle)  behaves  when  it  is  acted  upon  by  an  unbalanced  force, 
and  the  natural  habit  of  thinking-in-terms-of-human-values 
shows  itself  after  the  most  exacting  class-room  drill.  Instead 
of  giving  his  attention  narrowly  to  what  is  taking  place  at  an 
instant,  the  student,  in  his  groping  for  human  values  (which  do 
not  exist  in  the  bare  elements  of  a  subject),  is  pretty  sure  to 
refer  to  past  history  and  to  future  prospects;  his  idea  as  to 
what  is  taking  place  is  apt  to  remain  widely  inclusive,  as  if  he 
were  thinking  of  a  complex  human  experience  like  two  hours  at 
a  play,  or  a  month's  vacation,  or  four  years  of  war.  Very  often 
the  homely  wording  of  the  question  as  stated  above  betrays  the 
student  into  a  disclosure  of  his  native  contempt  for  precise  ideas 
and  he  answers  naively  that  "the  body  moves  in  the  direction 
of  the  force."  It  is  very  difficult  to  develop  the  habit  of  analyt- 
ical thinking. 

Our  teachers  of  mathematics  seem  to  know  that  they  are 
responsible  for  a  certain  kind  of  training,  and  they  are  not 
ashamed  to  apply  themselves  narrowly  in  its  accomplishment. 


vi  CALENDAR  OF  LEADING  EXPERIMENTS. 

Indeed,  many  of  us  who  do  not  share  their  responsibility  are  inclined 
to  think  that  mathematics  teachers  apply  themselves  too  nar- 
rowly, because  their  students,  as  we  get  them,  are  deficient  in 
mathematical  ideas  although  strong,  it  maybe,  on  forms. 

Physics  teachers  also  have  a  definite  responsibility,  and  our 
notion  as  to  this  responsibility  is  suggested  by  examples  a 
and  b  above.  The  physics  teacher,  however,  is  strongly  imbued 
with  the  idea  that  his  function  is  to  disclose  to  the  student  his 
(the  physicist's)  "domain  of  nature,"  and  he  is  inclined  there- 
fore to  discuss  results — Helmholtz's  theory  of  the  origin  of  the 
sun's  heat,  Lord  Kelvin's  calculations  of  the  age  of  the  earth, 
and  many  other  things  of  equal  speculative  interest — but  such 
things  prematurely  considered  divert  the  student's  attention 
away  from  elementary  or  irreducible  ideas  and  conceptions  with- 
out which  no  analytical  thinking  is  possible. 

Results  are  fine  inventions 

For  gentlemen  who  see; 
But  the  micro-scope  is  needed 

In  this  emergency. 


Primarily  this  book  has  to  do  with  class-room  experiments  in 
physics.  The  best  experiments  are  those  that  are  homely  and 
simple,  and  suggestive  rather  than  informing.  The  physics 
lecturer  should  pull  ideas  out  of  things  like  a  prestidigitateur, 
and  many  of  his  demonstrations  should  be  mere  motion  experi- 
ments,* by  which  we  do  not  mean  experiments  which  have  to  do 
with  the  theory  of  motion!  We  do  not  believe  in  the  lecture 
experiment  which  aims  at  a  numerical  result  rather  than  at  an 
idea, — decidedly  we  do  not;  and  our  pet  aversion  is  the  "study  in 
still  life,"  if  we  may  so  describe  the  deadly  lantern  slide.  Imag- 
ine a  sleight-of-hand  performer  making  use  of  lantern  slides! 
It  is  unthinkable,  and  yet  there  never  was  a  sleight-of-hand 
performer  who  had  one  tenth  of  the  resources  of  the  physics 
lecturer  in  college.  In  a  two-years'  course  in  elementary  physics 

*  See  pages  51  and  82  for  examples. 


PREFACE.  vii 

we  use,  by  actual  count,  31  lantern  slides;  and  out  of  a  total  of 
48  lecture  periods  in  the  two  years'  course  we  have  always  used  8 
or  more  for  written  tests  on  text-book  and  recitation  work.  This 
may  seem  to  some  as  a  damning  confession,  but  the  constraint 
which  leads  to  analytical  thinking  cannot  be  made  to  operate 
through  any  combination  of  talk  and  show.  No  Missourian  was 
ever  regenerated  by  talk;  and  even  when  you  have  shown  him  he 
remains  what  he  was  before — a  Missourian. 

Secondarily  this  book  is  intended  to  set  forth  the  possibilities 
of  an  extended  course  in  elementary  dynamics,  including  the 
dynamics  of  wave  motion. 


Some  things  in  this  book  may  seem  to  call  for  apology,  for  we 
simply  cannot  refrain  from  an  occasional  diversion  in  the  way  of 
poking  fun,  because  so  many  things  in  teaching  are  funny,  from 
our  point  of  view. 

From  our  point  of  view.  Let  no  one  who  reads  this  book  lose 
sight  of  the  qualification.  We  might,  of  course,  have  dwelt 
here  and  there  on  faults  of  our  own,  but  such  things  are  not  funny 
— from  our  point  of  view.  For  example,  we  have  many  times 
caught  ourselves  mistaking  the  fixity  of  an  idea  for  its  raison 
detre,  whereas  the  fixity  of  an  idea  is  not  the  same  thing  as 
reason,  especially  when  it  comes  to  making  the  idea  clear  to  a 
young  student.  Also  we  have  many  times  been  victims  of  the 
"illusion  of  activity"  by  which  we  mean  the  sense  of  one's 
effectiveness  which  comes  from  being  wholly  engaged  in  an 
undertaking.  "The  lecture  we  gave  this  morning ;  how  complete 
and  perfect  it  was;  what  a  masterpiece  of  edification!"  The 
only  cure  for  this  illusion  is  the  reaction  of  the  students,  and  no 
teacher  who  seeks  and  uses  this  cure  can  be  proud  of  his  work; 
although  he  may  develop  a  humble  but  invincible  self-respect 

in  that  he  never  fails  to  do  his  best. 

W.  S.  FRANKLIN, 

BARRY  MAcNurr, 

October  27,  1917. 


TABLE  OF  CONTENTS. 


PAGES 

PART  I.     Mechanics 1-68 

PART  II.     Heat 69-94 

PART  III.     Electricity  and  Magnetism 95-138 

PART  IV.    Light 139-156 

PART  V.    Sound 157-162 

PART  VI.     A  simple  treatise  on  wave  motion 163-205 

Appendix  A.     A  visitor's  laboratory  of  physics 207-210 

DISCONNECTED  ESSAYS. 

On  the  Study  of  Science 2 

Operative  and  inoperative  definitions 70 

The  side-stepping  of  mathematics , 96 

Bacon's  New  Engine 140 

The  philosophy  of  steam  shovels  and  the  philosophy  of  living  1 58 

Science  and  technology  versus  the  humanities  in  education.  164 
The  Traditive  Lamp,  or  the  proper  method  for  handing  down 

the  sciences  to  posterity 206 


PART  I. 
MECHANICS. 

A  very  simple  and  complete  treatment  of  the  elementary 
theory  of  elasticity  is  given  on  pages  182  to  218  of  Franklin  and 
MacNutt's  Mechanics  and  Heat,  The  Macmillan  Co.,  New  York 
City,  1910;  and  some  fundamental  experimental  demonstrations 
in  elasticity  are  described  by  Franklin  and  MacNutt  on  pages 
90-100  of  the  Bulletin  of  the  Society  for  the  Promotion  of  Engineer- 
ing Education,  October,  1916. 


THE   STUDY   OF   SCIENCE. 

Everyone  knows  of  the  constraint  which  is  placed  upon  men  by  the  physical 
necessities  of  the  world  in  which  we  live,  and  although  in  one  way  this  constraint  is 
more  and  more  relieved  with  the  advancement  of  the  sciences,  in  another  way  it 
becomes  more  and  more  exacting.  It  is  indeed  easier  to  cross  the  Atlantic  ocean 
now  than  it  was  in  Lief  Ericsson's  time;  but  think  of  the  discipline  of  the  present 
day  shop  and  consider  the  rules  of  machine  design,  the  rigors  of  the  mathematical 
sciences!  Could  even  the  hardy  Norsemen  have  known  anything  as  uncom- 
promisingly exacting  as  these? 

It  is  no  wonder  that  easy-going  believers  in  liberal  education  have  always 
looked  with  horror  on  the  sciences,  very  much  as  softened  men  and  women  look 
upon  work.  Liberalism  means  freedom,  and  "liberalism  in  education  is  the  freedom 
of  development  in  each  individual  of  that  character  and  personality  which  is  his 
true  nature."  All  this  we  accept  in  a  spirit  of  optimism,  believing  men's  true 
natures  to  be  good;  but  there  is  a  phase  of  education  which  has  but  little  to  do, 
directly,  with  character  and  personality,  and  we  call  to  your  attention  this  concep- 
tion of  liberalism  in  education  in  order  that  we  may  turn  sharply  away  from  it 
as  an  incomplete  conception  which  to  a  great  extent  excludes  the  mathematical 
sciences.  Indeed  we  wish  to  point  out  a  condition  in  education  which  is  the  anti- 
thesis of  freedom;  for  the  study  of  the  mathematical  sciences  means  a  reorganiza- 
tion of  the  work-a-day  mind  of  a  young  man  as  complete  in  its  sphere  as  the  pupation 
of  an  insect,  and  an  exacting  constraint  is  the  essential  condition  of  this  reorganiza- 
tion. 

This  statement  is  taken  from  The  Study  of  Science,  an  introduction  to  Franklin 
and  MacNutt's  Mechanics  and  Heat,  The  Macmillan  Co.,  1910.  This  essay  is 
reprinted  in  Bill's  School  and  Mine,  A  Collection  of  Essays  on  Education,  by  W.  S. 
Franklin,  published  by  Franklin,  MacNutt  and  Charles,  South  Bethlehem,  Pa. 
In  this  reprint  it  is  stated  that  the  essay  "  is  a  sticker,  and  that  any  particular  reader 
who  does  not  like  it  can  leave  it  alone."  But  there  is  an  increasing  number  of 
young  men  in  this  world  who  must  study  science  whether  they  like  it  or  not;  and 
this  essay  is  intended  to  explain  this  remarkable  and  in  many  respects  distressing 
fact. 


MASS  AND  WEIGHT. 

"  In  defense  of  accuracy  we  must  be  zealous,  as  it 
were,  even  to  slaying." — P.  G.  Tait. 

Many  of  our  teachers  of  physics  and  engineering  are  lacking 
in  complete  precision  of  thought  concerning  elementary  mech- 
anics largely  because  they  refuse  to  think  of  the  fundamental 
ideas  of  mechanics  with  a  greater  degree  of  mathematical  or 
logical  precision  than  seems  to  them  to  be  required  by  the  degree 
of  measuring  precision  that  satisfies  the  research  physicist  and 
the  engineer;  but  precision  of  thought  is  not  dependent  upon 
precision  of  measurement.  Indeed  precision  of  measurement  in 
modern  physics  is  mostly  a  tradition  which  we  have  inherited 
from  spherical  astronomy;  precision  of  thought  is  vastly  more 
important. 

The  term  mass  means  quantity  of  material  as  measured  by  a 
balance  scale,  and  it  is  properly  expressed  in  grams  or  pounds. 
In  commerce,  however,  we  speak  of  the  weight  of  a  batch  of 
sugar  or  coal  as  weighed  by  a  balance  scale,  and  the  word  weight 
so  used  means  exactly  the  same  thing  as  mass. 

When  we  speak  of  the  weight  of  a  body  in  this  text  we  mean 
the  force  with  which  the  earth  pulls  on  the  body,  and  it  is  properly 
expressed  in  dynes  or  poundals.  The  pull  of  the  earth  on  a 
one-pound  body  in  London  is  a  definite  force,  it  is  extensively 
used  as  a  unit  of  force,  and  it  is  called  the  "pound."  The 
weight  of  a  body,  that  is  to  say,  the  earth-pull  on  a  body  may  be 
properly  expressed  in  "pounds"  but  the  coal  man's  scales  do  not 
determine  the  weight  of  a  body  in  "pounds" 

One  "pound"  of  force  gives  an  acceleration  of  32.174  feet 
per  second  per  second  to  a  mass  of  one  pound,  so  that,  according 
to  Newton's  second  law,  one  "pound"  of  force  will  give  an 
acceleration  of  one  foot  per  second  per  second  to  32.174  pounds 

3 


4  CALENDAR  OF  LEADING  EXPERIMENTS. 

of  material  (one  slug  of  material) .  Therefore  if  force  is  expressed 
in  "pounds,"  mass  in  slugs  and  acceleration  in  feet  per  second 
per  second  the  simple  form  of  the  equation  F  =  ma  can  be 
used. 

To  reduce  pounds  of  coal  as  measured  by  a  balance  scale  to 
slugs,  divide  by  32.174  (a  pure  number,  not  an  acceleration). 
The  weight  W  of  a  body  (the  force  with  which  the  earth  pulls 
a  body)  is  of  course  equal  to  mg,  so  that  the  mass  of  the  body 
m  equals  W/g.  Thus  the  actual  local  weight  of  a  body  in 
London  "pounds1'  divided  by  the  local  value  of  g  gives  the 
mass  of  the  body  in  slugs*;  but  when  your  coal  man  sends  you 
a  bill  for  what  he  calls  a  certain  "weight"  of  coal,  say  2,000 
pounds,  do  not  divide  this  kind  of  "weight"  by  g  to  get  mass! 

The  system  of  mechanical  units  which  involves  the  "pound" 
as  a  unit  of  force  is  very  widely  and  properly  used,  and  every 
teacher  of  elementary  mechanics  should  use  this  system  of  units 
unhesitatingly.  We  call  the  system  the  foot-slug-second  system 
for  obvious  reasons.  There  is  no  objection  to  the  precisef  use 
of  the  foot-slug-second  system  except  on  the  ridiculous  assump- 
tion that  the  primary  business  of  the  teacher  is  to  promote  the 
metric  system ;  but  it  is  improper  for  anyone  to  agree  to  use  the 
word  weight  for  the  force  with  which  the  earth  pulls  on  a  body 
and  then  carelessly  revert  to  the  usage  to  the  grocer  and  the 
coal  man. 

The  weight  of  a  body  in  London  in  "pounds"  is,  of  course, 
equal  to  its  mass  in  pounds,  and  it  has,  more  than  once,  been 
proposed  to  call  the  weight  of  a  body  in  London  its  standard 

*  A  perplexing  mix-up  of  dimensions  is  involved  in  the  double  meaning  of  the 
word  pound.  Out  of  respect  for  etymology  we  may  postulate  identical  dimen- 
sions to  the  pound  of  sugar  and  the  "pound"  of  push  or  pull,  or  we  may  postulate 
identical  dimensions  to  pounds  of  sugar  and  slugs  of  sugar.  We  prefer  the  second 
postulate,  and,  in  accordance  with  this  postulate,  one  must  divide  pounds  of  sugar 
by  32.174  (a  numeric)  to  get  slugs  of  sugar,  and  one  must  divide  the  local  weight 
of  a  body  in  London  "pounds"  by  the  local  acceleration  of  gravity  (a  denominate 
number)  to  get  mass  in  slugs.  From  this  point  of  view  the  "pound"  of  force 
has  the  same  dimensions  as  the  poundal  of  force. 

t  See  page  50  of  this  volume  for  further  discussion  of  this  subject. 


MECHANICS.  5 

weight  and  use  the  term  standard  weight  instead  of  mass;  but 
there  are  two  objections  to  this,  namely,  (a)  The  term  mass  is 
almost  universally  recognized  by  physicists  and  chemists,  and 
(b)  The  mass  of  a  given  body  is  independent*  of  time  and  place, 
it  has  to  do  only  with  an  invariant*  relation  between  the  given 
body  and  the  standard  kilogram  (a  piece  of  metal),  and  extra- 
neous and  confusing  ideas  would  be  introduced  by  the  use  of  the 
term  standard  weight  because  this  term  implies  location  and  a 
relationship  between  the  given  body  and  the  earth.  How  awk- 
ward it  would  be,  for  example,  to  be  obliged  always  to  speak  and 
think  of  the  distance  d  between  two  points  (x,  y,  z)  and  (#',  y' ', 
*')  as  [(x  -  x')*  +  (y  -  y')*  +  (z  -  z')2]1/2.  This  is  an-  in- 
variant and  the  most  useful  name  or  symbol  for  it  is  a  name 
or  symbol  which  carries  no  redundant  suggestions  as  to  particular 
axes  of  reference,  and  this  would  be  true  even  if  we  had  always  to 
make  use  of  a  particular  set  of  reference  axes  in  the  measurement 
of  d. 

Many  engineering  writers  pretend  never  to  use  the  poundal  as 
a  unit  of  force,  and  we  use  the  poundal  chiefly  as  a  name  for  a 
step  in  an  argument.  It  is  better  to  name  it  than  to  refrain  from 
naming  it,  thereby  pretending  not  to  use  it.  Everyone  who 
expresses  mass  in  pounds  does  use  the  poundal  as  a  unit  of  force, 
and  every  English-speaking  person  expresses  mass  in  pounds 
when  he  buys  sugar  or  coal.  If  a  man  steals  sugar  no  accounting 
is  necessary,  at  leas  no  book-keeping  is  necessary,  and  only 
such  men  among  English-speaking  people  can  honestly  claim  not 
to  use  the  poundal  as  a  unit  of  force!  The  question  would 
appear  therefore  to  be  a  question  of  honesty  versus  dishonesty, 
but  it  is  not  quite  as  bad  as  that,  for  the  second  horn  of  the 
dilemma  is  unconscious  dishonesty  or  sophistry,  and  this  un- 
conscious dishonesty  has  been  carried  to  the  limit  by  a  recent 
writer  who  speaks  of  the  coal  man's  unit  of  mass  as  scholastic. f 

*  No  consideration  is  here  given  to  variations  of  mass  as  recognized  in  the 
recent  developments  of  the  principle  of  relativity. 

t  Professor  E.  V.  Huntington,  of  Harvard  University.  See  American  Mathe- 
matical Monthly,  January,  1917,  page  16.  In  his  recent  articles  on  the  fundamentals 


6  CALENDAR  OF   LEADING   EXPERIMENTS. 

of  dynamics  Professor  Huntington  has  not  contributed  anything  to  the  logical  or 
mathematical  aspects  of  the  subject  which  has  not  been  known  in  all  simplicity 
and  completeness  since  the  days  of  Thomson  and  Tait,  he  is  definitely  astray 
logically  in  his  attempt  to  ignore  the  fundamental  idea  of  mass  (see  page  10 
of  this  volume),  and  he  shows  a  peculiar  lack  of  appreciation  of  the  constraining 
exactions  which  are  placed  upon  the  physicist  by  measuring  operations.  For 
example,  Professor  Huntington  says,  with  easy  assurance  that,  "fundamental 
units  may  be  chosen  at  pleasure,"  whereas  the  choice  of  fundamental  units  is 
governed  by  practical  considerations;  in  the  first  place  the  fundamental  units 
must  be  preserved  as  material  standards,  and  in  the  second  place  the  fundamental 
quantities  must  be  susceptible  of  very  accurate  measurement  because  the  definition  of 
a  derived  unit  cannot  be  realized  with  greater  accuracy  than  the  fundamental  quantities 
can  be  measured. 

Think  of  the  years  of  confusion  in  electrical  measurements  when  the  theoretical 
ohm  could  not  be  produced  with  greater  accuracy  than  one  or  two  per  cent  and 
when  anybody  could  make  resistance  measurements  to  one  tenth  of  a  per  cent! 
When  we  think  of  this  old  nightmare  and  read  Professor  Huntington 's  argument 
for  the  adoption  of  force  as  a  fundamental  quantity,  what  do  you  suppose  we  are 
inclined  to  say?  We  refrain  from  saying  it,  at  any  rate.  Professor  Huntington 
assures  us  that  a  spring  can  be  easily  preserved,  whereas  the  best  and  most  care- 
fully "aged"  measuring  springs  in  existence  grow  perceptibly  softer  and  softer  in 
use.  Tempered  steel  is  in  a  meta-stable  condition  and  so  also  is  hardened  phosphor 
bronze  and  fused  quartz.  Of  course  the  "pound"  of  force,  the  pull  of  the  earth 
on  the  standard  pound  in  London,  is  adequately  preserved,  but  Professor  Hunting- 
ton  seems  nevertheless  to  think  that  a  unit  of  force  must  be  preserved  in  the  form 
of  a  stretched  spring! 

One  of  the  very  few  redeeming  features  of  the  recent,  long-drawn-out  rehash  of 
elementary  dynamics  in  Science  and  in  the  American  Mathematical  Monthly  is  the 
article  by  Professor  L.  M.  Hoskins  of  Stanford  University,  Science,  April  23,  1915, 
pages  608-611.  The  rehash  is  due  primarily  to  Professor  Huntington,  and  its 
significance  lies  almost  solely  in  its  showing  that  nearly  everybody  seems  to  take 
anybody  seriously  when  anybody  happens  to  be  a  Harvard  professor.  See  page 
34  of  this  volume  for  further  comment. 


ON   UNBALANCED   FORCE. 

When  a  car  is  being  started  by  a  locomotive  the  forward  pull 
of  the  locomotive  on  the  car  is  greater  than  the  backward  pull 
or  drag  of  track  and  wind  on  the  car;  all  the  forces  which  act 
on  the  car  are  together  equivalent  to  a  single  force  pulling  for- 
wards on  the  car,  and  this  single  force  constitutes  what  is  called 
an  unbalanced  force,  Whenever  the  velocity  of  a  body  is  chang- 
ing the  body  is  being  acted  upon  by  an  unbalanced  force.  As 
an  example  let  us  consider  a  ball  which  is  tied  to  a  string  and 
twirled  in  a  circle.  All  the  forces  acting  on  the  ball,  namely,  the 
gravity  pull  of  the  earth,  the  drag  of  the  air  and  the  pull  of  the 
string,  are  together  equivalent  to  a  single  force. 

One  of  the  most  frequent  perplexities  of  the  student  in  regard 
to  unbalanced  force  comes  from  a  wrong  interpretation  of  the 
principle  of  equality  of  action  and  reaction.  In  the  above 
example  the  string  pulls  inwards  on  the  ball  and  the  ball  pulls 
outwards  on  the  string.  We  refer  here  to  the  two  forces  acting 
at  the  point  where  the  string  is  attached  to  the  ball,  and  these 
two  forces  act  on  different  bodies  as  stated,  namely,  the  string 
pulls  on  the  ball  and  the  ball  pulls  on  the  string.  Two  such 
forces  which  act  on  different  bodies  have  nothing  whatever  to 
do  with  the  question  as  to  whether  the  forces  which  act  on  a  given 
body  are  balanced  or  unbalanced.  This  matter  is  discussed  more 
at  length  on  page  12  of  General  Physics. 

The  misunderstanding  and  false  application  of  the  principle 
of  equality  of  action  and  reaction  is  so  common  that  every  teacher 
of  physics  should  make  a  searching  analysis  of  the  matter; 
he  should  not  accept  anyone  as  authority  but  he  should  think 
for  himself  and  he  should  by  all  means  pay  attention  to  simple 
essentials.  Do  you  agree  as  to  what  is  said  above  concerning 
unbalanced  force?  Do  you  agree  that  action  and  reaction  are 
equal  and  opposite  forces  acting  on  different  bodies  and  there- 

7 


8  CALENDAR  OF   LEADING   EXPERIMENTS. 

fore  having  nothing  whatever,  directly  or  indirectly,  to  do  with 
the  question  as  to  whether  the  forces  which  act  on  a  given  body 
are  balanced  or  unbalanced?  If  you  are  doubtful  consider  a 
trade  between  two  persons  A  and  B.  This  trade  is  a  purchase 
from  A's  point  of  view  and  a  sale  from  B's  point  of  view,  and 
of  course  the  sale  aspect  of  this  mutual  operation  is  equal  to  its 
purchase  aspect;  but  everyone  understands  that  a  given  person 
may  buy  more  than  he  sells  or  sell  more  than  he  buys.  A  trade 
is  a  single  thing,  and  yet  it  is  very  convenient  to  speak  of  a  trade 
as  a  sale  or  purchase  in  its  relation  to  a  given  person.  The  force 
action  between  two  bodies  A  and  B,  involving  the  exertion 
of  equal  and  opposite  forces  on  the  two  bodies,  is  a  single  thing, 
a  stress,  let  us  say,  and  yet  it  is  convenient  to  speak  of  a  stress 
as  a  force  acting  on  A  or  as  an  equal  and  opposite  force  acting 
on  B.  Indeed  it  is  logically  necessary  to  think  of  one  aspect 
of  a  stress  when  considering  a  given  body.  Iowa  is  north  of 
Missouri — yes  but  Missouri  is  south  of  Iowa!  He  who  would 
argue  that  an  unbalanced  force  action  on  a  given  body  cannot 
be  because  action  and  reaction  are  always  exactly  equal  and 
opposite  is  like  the  purist  who  would  wish  to  invent  a  fancy  way 
of  saying  that  Missouri  is  south  of  Iowa,  fearing  a  verbal  battle 
with  the  man  from  Missouri  who  knows  that  Iowa  is  north  of 
Missouri!  Imagine  a  man  who,  in  considering  that  A  is  north 
of  B  and  that  therefore  B  is  the  same  distance  south  of  A, 
concludes  that  nothing  can  really  be  north  or  south  of  anything 
else.  The  geometric  sense  of  such  a  man  would  be  on  a  par 
with  the  mechanical  sense  of  one  who  concludes  that  an  un- 
balanced force  cannot  exist  because  action  and  reaction  are 
always  equal  and  opposite  and  on  a  par  with  the  business  sense 
of  a  merchant  who  would  conclude  from  the  equal  and  opposite 
aspects  of  a  trade  that  it  would  be  impossible  to  buy  more  than 
he  sells! 

All  the  forces  which  act  on  a  given  body  are  often  equivalent  to  a 
single  resultant  unbalanced  force,  and  it  is  such  a  force  that  changes 
the  velocity  of  the  body. 


MECHANICS.  9 

The  recently  developed  electromagnetic  theory  (including  the 
theory  of  relativity)  does  not  vitiate  this  point  of  view  in  the 
least,  although  it  does  modify  our  notions  of  mass  and  point  to 
minute  (ordinarily  extremely  minute)  force  actions  between 
accelerated  bodies  and  the  ether  of  space  if  we  may  use  that  term 
merely  to  designate  widespread  associated  energy.  Indeed  the 
principle  of  relativity  (for  it  should  no  longer  be  called  a  theory) 
does  not  vitiate  the  principle  of  equality  of  action  and  reaction 
if  due  attention  be  given  to  these  etherial  force  actions. 

A  "body"  includes  everything  inside  of  a  certain  bounding 
surface,  and  at  moderate  velocities  and  accelerations  only  an 
extremely  minute  fraction  of  the  kinetic  energy  and  momentum 
of  a  body  resides  outside  of  this  boundary,  according '  to  the 
electromagnetic  theory,  and  only  an  extremely  minute  fraction 
of  an  accelerating  force  is  balanced  by  force  action  across  this 
boundary;*  and  a  recent  attemptf  to  broaden  the  science  of 
dynamics  by  postulating  an  ether  force  which  completely  balances 
every  accelerating  force  is  unjustifiable  and  indeed  meaningless. 

*  The  integral  of  the  well-known  Maxwellian  stress  across  the  bounding  surface. 

f  Professor  H.  M.  Dadourian's  Analytical  Mechanics.  A  very  sane  review  of 
this  book  by  Professor  E.  W.  Rettger  is  given  in  Science,  January  23,  1914,  pages 
140-142.  See  page  34  of  this  volume  for  further  comment. 


THE   FUNDAMENTAL   EQUATIONS   OF   DYNAMICS. 


The  recent  attempt*  to  rule  out  mass  as  a  fundamental  con- 
cept in  dynamics  makes  it  advisable  to  supplement  what  is 
given  in  General  .Physics  on  the  fundamental  equations  of 
dynamics  (see  pages  8— 10).  We  are  here  concerned  with  the 
acceleration  of  a  body  when  acted  upon  by  an  unbalanced  force. 
The  acceleration  varies  from  body  to  body  for  a  given  identifiable 
force,  and  it  varies  from  force  to  force  for  a  given  body.  These 
are  two  equally  fundamental  modes  of  variation  and  both  of 
them  must  be  formulated  in  the  fundamental  equation  or 
equations  of  dynamics,  and  of  course  this  formulation  must  be 
based  on  experiment. 

Given  three  bodies  A,  B  and  C,  and  three  identifiable 
forces  a,  b  and  c.  Let  the  acceleration  of  each  body  due  to 
each  force  be  observed,  the  results  being  as  shown  in  the  accom- 
panying table. 

TABLE  OF  OBSERVED  ACCELERATIONS. 

Bodies 


Forces 


A 

B 

C 

a 

25 

30 

35 

b 

50 

60 

70 

c 

75 

90 

105 

*  By  Professor  E.  V.  Huntington  who  says  that  equation  (2)  below  is  a  mathe- 
matical consequence  of  equation  (i) !  See  Science  March  3,  1916,  page  315. 

Equation  (i)  expresses  the  relation  between  the  accelerations  a  and  a'  of  a  given 
body  due  to  two  forces  F  and  F',  respectively,  and  pure  logic  would  not  know  of 
even  the  existence  of  any  other  body.  See  page  34  of  this  volume  for  further 
comment. 

10 


MECHANICS.  II 

Let  us  suppose  that  the  table  has  been  extended  so  as  to  include 
a  great  many  different  forces  and  a  great  many  different  bodies, 
then  a  careful  inspection  of  the  table  would  lead  to  the  following 
generalizations. 

(a)  If  one  force  produces  twice  as  much  acceleration  as  another 
force  when  acting  on  a  given  body,  then  the  one  force  produces 
twice  as  much  acceleration  as  the  other  force  when  acting  on 
any  body  whatever. 

(b)  If  one  body  is  accelerated  twice  as  much  as  another  body 
under  the  action  of  a  given  force,  then  the  one  body  is  accelerated 
twice  as  much  as  the  other  body  under  the  action  of  any  force 
whatever. 

The  experimental  fact  (a)  makes  it  convenient  to  define  the 
ratio  of  two  forces  as  the  ratio  of  the  accelerations  they  produce 
when  acting  on  a  given  body,  because  this  ratio  is  the  same  for 
all  bodies.  That  is 

Fa 


where  a  is  the  acceleration  of  a  given  body  produced  by  force 
Fj  and  a!  is  the  acceleration  produced  by  force  Ff. 

The  experimental  fact  (b)  makes  it  convenient  to  define  the 
ratio  of  the  masses  of  two  bodies  as  the  inverse  ratio  of  the 
accelerations  produced  by  a  given  force,  because  this  ratio  is 
the  same  for  all  forces.  That  is 

m       a' 

where  a  is  the  acceleration  of  body  No.  I  and  a'  is  the  accelera- 
tion of  body  No.  2  both  produced  by  a  given  force,  and  m  and 
m'  are  the  masses  of  the  respective  bodies. 

Equations  (i)  and  (2)  are  the  fundamental  equations  of 
dynamics. 

Experiment  shows  that  the  ratio  mfm'  as  defined  by  equation 
(2)  is  exactly  equal  to  the  ratio  of  the  masses  of  the  two  bodies 
as  measured  on  a  balance  scale,  and  since  the  balance  scale 


12  CALENDAR  OF   LEADING   EXPERIMENTS. 

measures  mass  very  conveniently  and  with  extreme  precision 
it  is  best  to  define  mass  as  measured  by  the  balance  scale  and 
accept  equation  (2)  as  an  experimental  discovery. 

From  equations  (i)  and  (2)  it  is  evident  that  the  acceleration 
of  a  body  is  proportional  to  the  accelerating  force  and  inversely 
proportional  to  the  mass  of  the  body. 

Experiment  I. — The  authors  have  found  it  best  to  use  only 
extremely  simple  lecture  experiments  in  the  discussion  of  the 
laws  of  motion.  Taking  a  box  in  the  hands,  call  attention  to 
the  fact  that  you  have  to  exert  an  upward  force  on  the  box  to 
balance  the  downward  pull  of  the  earth.  Call  on  a  young  man 
to  pull  on  the  box,  and  point  out  that  in  addition  to  supporting 
the  box  against  gravity  you  have  to  balance  the  force  exerted 
by  the  young  man  to  keep  the  box  stationary. 

Point  out  that  everything  as  stated  applies  to  a  box  similarly 
held  in  a  steadily  moving  car  or  boat  on  a  straight  track  or  course, 
so  that  the  forces  which  act  on  a  body  are  balanced  when  the 
body  is  stationary  or  moving  uniformly  along  a  straight  path 
but  surrounded  on  all  sides  by  things  moving  along  with  it. 

How  about  a  body  which  moves  steadily  along  a  straight 
path  but  not  surrounded  by  bodies  which  move  along  with  it? 
Everyone  knows  that  an  active  agent  such  as  a  horse  or  a  steam 
engine  must  pull  steadily  on  such  a  body  to  keep  it  in  motion. 
If  left  to  itself  such  a  moving  body  quickly  conies  to  rest.  This 
tendency  for  a  moving  body  to  come  to  rest  is  due  to  dragging 
forces  or  friction  exerted  on  the  moving  body  by  surrounding 
bodies.  Thus  a  moving  boat  is  brought  to  rest  by  the  drag  of 
the  water  when  the  propelling  force  ceases  to  act ;  a  train  of  cars 
is  brought  to  rest  by  a  frictional  drag  when  the  pull  of  the 
locomotive  ceases;  a  box  which  is  drawn  steadily  across  the 
table  comes  to  rest  when  left  to  itself  because  of  the  dragging 
force  due  to  friction  between  the  moving  box  and  the  table. 
We  must,  therefore,  always  consider  two  forces  when  we  think 
of  a  body  which  is  kept  in  motion  like  a  car  or  boat,  namely,  the 
propelling  force  due  to  some  active  agent  and  the  dragging  force 


MECHANICS.  13 

of  friction.  Newton  pointed  out  that  when  a  body  is  moving 
steadily  along  a  straight  path  the  propelling  force  is  equal  and 
opposite  to  the  dragging  force  of  friction.  See  General  Physics, 
Art.  3,  page  7. 

Let  us  now  consider  the  force*  which  must  act  on  a  body 
which  is  changing  its  velocity,  upon  a  body  which  is  being  started 
or  stopped,  for  example.  Every  one  has  noticed  how  a  mule 
strains  at  his  rope  when  starting  a  canal  boat,  especially  if  the 
boat  is  heavily  loaded,  and  how  the  boat  continues  to  move  for 
a  long  time  after  the  mule  ceases  to  pull.  In  the  first  case  the 
pull  of  the  mule  greatly  exceeds  the  backward  drag  of  the  water, 
and  the  velocity  of  the  boat  increases;  in  the  second  case  the 
backward  drag  of  the  water  exceeds  the  pull  of  the  mule,  for  the 
mule  is  not  pulling  at  all,  and  the  velocity  of  the  boat  decreases. 
When  the  velocity  of  a  body  is  changing  the  forces  which  act 
on  the  body  are  unbalanced.  We  may  therefore  conclude  that 
the  effect  of  an  unbalanced  force  on  a  body  is  to  change  the 
velocity  of  the  body,  and  it  is  evident  that  the  longer  the  un- 
balanced force  continues  to  act  the  greater  the  change  of  velocity. 
Thus  if  the  mule  ceases  to  pull  on  a  canal  boat  for  one  second 
the  velocity  of  the  boat  will  be  but  slightly  reduced  by  the 
unbalanced  backward  drag  of  the  water,  whereas  if  the  mule 
does  not  pull  for  ten  seconds  the  velocity  of  the  boat  will  be 
reduced  to  a  much  greater  .extent.  In  fact  the  change  of  velocity 
of  a  body  due  to  a  given  unbalanced  force  is  proportional  to  the 
time  that  the  force  continues  to  act.  This  is  exemplified  by  a  body 
falling  freely  under  the  action  of  the  unbalanced  pull  of  the 
earth  on  the  body;  during  one  second  the  body  gains  about  32 
feet  per  second  of  velocity,  during  two  seconds  it  gains  twice  as 
much  velocity  (about  64  feet  per  second),  and  so  on. 

Everyone  knows  what  it  means  to  give  an  easy  pull  or  a  hard 
pull  on  a  body.  That  is  to  say,  we  all  have  an  idea  that  a  force 
may  be  large  or  small.  Everyone  knows  also  that  under  the 

*  If  several  forces  act  we  here  refer  to  the  single  force  which  is  equivalent  to 
them  all.  Translatory  motion,  only,  is  here  considered. 


CALENDAR  OF  LEADING  EXPERIMENTS. 


action  of  a  hard  pull  a  canal  boat  will  get  under  way  more  quickly 
(gain  velocity  faster)  than  it  will  under  the  action  of  an  easy 
pull,  and  a  precise  statement  of  the  effect  of  an  unbalanced  force 
on  a  body  must  correlate  the  value  of  the  force  and  the  rate  at 
which  it  imparts  velocity  to  the  body.  This  seems  a  very  difficult 
thing,  but  its  difficulty  is  in  large  part  due  to  the  fact  that  we 
have  not  yet  agreed  as  to  what  we  mean  when  we  say  that  one 
force  is  exactly  three  or  four  or  any  number  of  times  as  large  as 
another  force.  Suppose  therefore  that  we  agree  to  call  one  force 
twice  as  large  as  another  when  it  will  produce  in  a  given  body  twice 
as  much  velocity  in  a  given  time  (remembering  that  we  are  talking 
about  unbalanced  forces).  As  a  result  of  this  definition  we  may 
state  that  the  amount  of  velocity  produced  per  second  in  a  given 
body  by  an  unbalanced  force  is  proportional  to  the  force. 

Note. — This  definition  of  the  ratio  of  two  forces  is  by  no 
means  sufficient  to  establish  the  equation  F/F'  =  a/a'.  A  very 
wide  range  of  experiment  is  necessary  to  show  that  this  equation 
is  true  for  all  bodies  if  it  is  true  for  one.  See  discussion  on 
pages  10  and  n  of  this  volume. 

2.  An  experiment  with  two  similar  blocks. — Drop  two  similar 
blocks  together  and  call  attention  to  the  fact  that  they  reach 


mass  m 


mass  gm 


yo\ 


force  F 


'acceleration  a 


mass  Z 


force  gf 


acceleration 


force  F 


Fig.  1. 


acceleration  a  \ 
I 

Fig.  2. 


Fig.  3. 


the  floor  at  the  same  time.  Two  blocks  fall  the  same  distance 
in  a  given  time,  therefore  they  fall  with  the  same  increasing 
velocity  and  with  the  same  acceleration  as  one  block.* 

*  The  friction  of  the  air  is  very  small  and  it  is  ignored. 


MECHANICS.  15 

The  mass  of  the  two  blocks  (as  measured  on  a  balance  scale) 
is  2w,  where  m  is  the  mass  of  one  block;  and  the  pull  of 
gravity  on  the  two  blocks  is  2F,  where  'F  is  the  pull  gravity 
on  one  block.  Therefore  when  m  and  F  are  both  doubled 
the  acceleration  a  remains  unchanged. 

This  argument  is  not  exactly  in  accordance  with  the  fundamental  equation  (i) 
on  page  ii  of  this  volume.  In  all  strictness  it  should  be  as  follows:  Let  us  adopt 
F/F'  =  a/a'  as  the  definition  of  the  ratio  of  two  forces,  and  let  us  assume  that  the 
pull  of  the  earth  on  the  two  blocks  (mass  2m)  is  twice  as  great  as  the  pull  of  the  earth 
on  one  block  (mass  m),  as  indicated  in  Fig.  2.  Imagine  the  force  in  Fig.  2  to  be 
reduced  to  half-value;  then  according  to  the  fundamental  equation  (FfF'  =  a/a') 
the  acceleration  would  be  reduced  to  %a,  as  indicated  in  Fig.  3.  Comparing 
Figs,  i  and  3  we  see  that  when  the  mass  of  a  body  is  doubled  a  given  force  F  pro- 
duces half  as  much  acceleration,  or  m/m'  =  a' /a,  which  is  equation  (2)  on  page 
ii  of  this  volume.  Therefore  the  above  assumption  is  justified,  that  is,  the  weight 
of  a  body  (pull  of  earth  on  the  body)  is  proportional  to  the  mass  of  the  body  as 
measured  by  a  balance  scale,  and  the  ratio  of  two  masses  as  defined  by  the  equation 
m/m'  =  a' fa  is  the  same  as  the  ratio  of  two  masses  as  measured  by  a  balance  scale. 

3.  Reaction  experiment. — An  electrically  driven  toy  loco- 
motive runs  on  a  circular  track  laid  near  the  edge  of  a  large 
horizontal  disk  which  is  supported  on  ball  bearings  so  as  to  turn 
freely  about  a  vertical  axis.  When  the  locomotive  starts  the  disk 
is  set  turning  backwards,  and  when  the  locomotive  stops  the 
disk  is  set  turning  forwards.  An  elevated  railway  structure  has 
to  be  braced  near  a  station  so  as  to  withstand  the  backward 
push  of  the  locomotive  as  it  starts  a  train  and  so  as  to  withstand 
the  very  great  forward  push  of  an  entire  train  which  is  brought 
to  rest  quickly  by  braking. 

Remark. — Is  the  kick  of  a  gun  the  reaction  which  corresponds 
to  the  push  of  the  powder  gases  on  the  projectile?  It  is  not. 
The  kick  of  the  gun  is  due  to  the  backward  push  of  the  powder 
gases  on  the  breech  plug  of  the  gun,  and  this  is  of  course  equal 
to  the  forward  push  of  the  breech  plug  on  the  powder  gases  (ac- 
tion is  equal  to  reaction  and  opposite  thereto) .  Also  the  forward 
push  of  the  powder  gases  on  the  projectile  is  equal  to  the  back- 
ward push  of  the  projectile  on  the  powder  gases  (action  is  equal 
to  reaction  and  opposite  thereto).  But  the  forward  push  of  the 


16  CALENDAR  OF  LEADING  EXPERIMENTS. 

breech  plug  on  the  powder  gases  is  not  equal  to  the  backward 
push  of  the  projectile  on  the  powder  gases;  the  powder  gases  are 
being  rapidly  accelerated  and  the  forces  which  act  on  the  powder 
gases  are  not  balanced. 


IMPULSE-VALUE  OF  A  FORCE. 

Consider  an  unbalanced  force  F  =  ma  acting  on  a  body,  and 
let  us  assume  F  to  be  constant.  Multiply  both  members 
of  this  equation  by  the  time  /  during  which  the  force  continues 
to  act,  and  we  have  Ft  =  mat.  But  at  (the  rate  of  gain  of  veloc- 
ity multiplied  by  elapsed  time)  is  the  total  velocity  v  produced 
by  the  force  F  during  the  time  t.  Therefore  we  have : 

Ft  =  mv 

The  product  mv  is  called  the  momentum  of  the  body,  and  the 
product  Ft  (the  value  of  a  force  multiplied  by  the  time  it  con- 
tinues to  act)  is  called  the  impulse-value  of  the  force. 

If  the  force      F      is  not  constant  we  have      F'dt  =  m—-dt  =  ni'dv,      and 

at 

jF'dt  =  fm-dv  =  mv.  The  integral  /F'dt  is  called  the  impulse  value  of  the  force 
F  and  it  is  equal  to  the  momentum  mv  which  is  produced. 

The  above  equation  applies  not  only  to  the  starting  of  a  body 
but  also  to  the  stopping  of  a  body.  For  example,  a  hammer  of 
mass  m  moving  at  velocity  v  strikes  an  obstacle  and  is  brought 
to  rest.  Let  F  be  the  average  force  with  which  the  obstacle 
pushes  backwards  against  the  hammer  while  stopping  it  (this  is, 
of  course,  equal  to  the  average  force  exerted  by  the  hammer  on 
the  obstacle),  and  let  /  be  the  time  during  which  this  average 
force  acts.  Then  F  =  ma,  where  F  is  the  average  backward 
force  exerted  on  the  hammer  (or  the  average  forward  force 
exerted  by  the  hammer)  and  a  is  the  average  rate  at  which 
the  hammer  is  losing  velocity.  But  during  time  /  the  ham- 
mer loses  all  of  its  velocity  so  that  at  =  v,  or  Ft  =  mat  =  mv, 
where  v  is  the  initial  velocity  of  the  hammer  (the  total  velocity 
lost  by  the  hammer).  Therefore  the  impulse- value  of  the  force 
exerted  by  the  hammer  is  equal  to  the  momentum  of  the  hammer. 
The  force  exerted  by  a  bullet  is  properly  expressed  in  terms  of  its 
impulse  value  (which  is  equal  to  the  momentum  mv  of  the  bullet). 
3  17 


l8  CALENDAR  OF   LEADING  EXPERIMENTS. 

4.  The  anvil  experiment. — A  heavy  anvil  rests  upon  a  yielding 
support,  and  yet  it  gives  a  satisfactory  base  upon  which  to  flatten  a 
piece  of  iron  by  a  hammer  blow.     The  enormous  force  exerted  by 
the  hammer  lasts  for  a  very  short  time,  say,  one  ten-thousandth 
of  a  second,  and,  although  the  anvil  is  set  in  motion  by  this 
force,  the  actual  distance  moved  by  the  anvil  in  the  ten-thous- 
andth part  of  a  second  is  very  small  and  entirely  negligible  from 
the  point  of  view  of  the  blacksmith.     The  anvil  continues  to 
move,  however,  long  after  the  hammer  blow,  and  as  it  con- 
tinues to  move  it  compresses  its  elastic  support  for,  say,  a  tenth 
of  a  second.     But  to  stop  the  anvil  in  a  tenth  of  a  second  the 
average  force  exerted  on  the  anvil  by  the  supporting  structure 
need  be  only  one  thousandth  as  great  as  the  average  value  of 
the  force  (exerted  by  the  hammer)  which  set  the  anvil  in  motion 
in  one  ten- thousandth  of  a  second. 

5.  The  coin  and  card  experiment. — Place  a  small  card  hori- 
zontally on  the  end  of  the  finger  and  on  top  of  the  card  place  a 
small  coin.     A  quick  thump  against  the  edge  of  the  card  drives  it 
out  from  under  the  coin,  and  the  coin  is  left  on  the  end  of  the 
finger.     During  the  very  short  time  required  for  the  card  to 
slide  out  from  under  the  coin,  the  card  exerts  a  forward  drag 
on  the  coin  and  imparts  to  the  coin  a  small  velocity;    but  the 
coin  has  time  to  travel  only  a  very  short  distance  before  the  card 
is  gone,  and  the  coin,  then  sliding  along  on  the  end  of  the  finger, 
is  very  soon  brought  to  rest  by  friction.     The  whole  distance 
moved  by  the  coin  while  being  started  by  the  forward  drag  of 
the  card  and  while  being  stopped  by  the  backward  drag  of  the 
finger  is,  perhaps,  one  hundredth  of  an  inch  or  less. 

This  coin-and-card  experiment  is  worth  while  if  it  is  carefully 
analyzed  as  above,  but,  like  most  paradoxical  experiments,  it  is 
worse  than  useless  if  not  accompanied  by  careful  analysis. 

6.  The  cord  paradox. — A  heavy  metal  ball    B    is  supported 
by  a  small  cord  c  as  shown  in  Fig.  4,  and  a  quick  hammer  blow, 
as  indicated,  breaks  the  heavy  cord    C.    The  hammer  causes  a 


MECHANICS. 


very  large  tension  in  cord  C  for  a  very  short  time  thus  setting 
the  ball  B  in  motion.  Then  the  ball  continues  to  move  for  a 
relatively  long  time,  stretching  cord  c. 
Thus  cord  c  has  a  very  long  time  in  which 
to  stop  the  ball  whereas  cord  C  had  an 
extremely  short  time  in  which  to  start  the 
ball.  Thus  if  cord  C  is  under  excessive 
tension  for  a  ten-thousandth  of  a  second, 
and  if  ball  B  thus  quickly  set  in  motion 
continues  to  move  downwards  for  a  tenth 
of  a  second  before  it  is  brought  to  rest  by 
the  increasing  tension  of  cord  c,  then  the 
average  excess  tension  in  cord  c  (in  ad- 
dition to  tension  due  to  the  steady  pull  of  the  earth  on  B) 
will  be  only  one  thousandth  as  large  as  the  average  tension  in 
cord  C  during  the  ten-thousandth  of  a  second. 


UNIFORMLY  ACCELERATED   TRANSLATORY 
MOTION. 

7.  Experiment  with  a  rolling  disk. — The  simplest  method  of 
obtaining  a  close  approximation  to  uniformly  accelerated  transla- 
tory  motion  with  small  acceleration  is  to  use  a  heavy  disk  with 
a  small  axle  rolling  down  an  inclined  track.     According  to  equa- 
tion  (ii)  on  page  23,  General  Physics,  this  disk  starting  from 
rest   (PI  =  o)   should  travel  a  distance  proportional  to  the  square 
of  the  elapsed  time.     Lay  off  from  the  starting  point  of  the 
rolling  disk  a  series  of  distances  proportional  to  I,  4,  9,  16,  25, 
etc.,  and  mark  the  points  so  located.     Then  as  the  disk  rolls 
down  the  inclined  track  it  will  mark  time  as  it  passes  these 
various  points.     In  order  that  this  experiment  may  be  effective, 
the  distances  should  be  chosen,  or  the  inclination  of  the  track 
should  be  adjusted  so  as  to  make  the  rolling  disk  mark  seconds 
as  it  passes  the  various  points,  and  the  accuracy  with  which 
the  disk  marks  seconds  may  be  appreciated  by  watching  the 
rolling  disk  and  listening  to  a  telegraph  sounder  operated  by  a 
contact  device  on  a  clock  pendulum. 

8.  Shooting  at  a  falling  ball. — When  the  initial  velocity   v\  of 
a  body  is  zero,  or  when  it  is  vertical,  we  have  the  ordinary  case 
of  a  falling  body.     In  this  case  equation  (ii)  on  page  23,  General 
Physics,  can  be  solved  by  simple  algebra,  and  all  calculations 
can  be  made  by  simple  arithmetic,  the  only  complication  being 
that  vi   is  to  be  considered  negative  when  it  is  upwards.     When 
the  initial  velocity    v\    is  not  vertical,  as  in  the  case  of  a  tossed 
ball,  the  falling  body  is  called  a  projectile.     In  this  case  the  entire 
argument  as  given  on  page  23,  General  Physics,  holds  good,  but 
geometric  addition  must  be  substituted  for  arithmetical  addition 
in  equation  (ii).     This  equation,  being  interpreted  as  shown  in 
Fig.  5,  means  that  the  position  A'  of  the  ball  after  /  seconds  is 

20 


MECHANICS. 


21 


found  by  adding  together  the  vectors   v\t  and   \gfi.     If  the  ball 

at   B   is  released  at  the  instant  the  projectile  leaves  the  muzzle 

of   the   gun   at    A ,    then   both 

balls,    A'    and    B',  will  always 

be  at   the     same    distance     \gfi 

from  the  line  AB.     Therefore  if 

the  gun  is  aimed   accurately  at 

the  point    B    the  two  balls  will 

always    come     together     at    a 

point     p. 

The  most  satisfactory  ar- 
rangement is  as  follows :  A  wooden  ball  has  a  hole  bored  through 
it  so  that  it  will  slip  over  a  half-inch  brass  tube,  and  a  sling  with 
rubber  bands  is  arranged  to  be  released  by  a  trigger  and  throw 
the  ball  along  the  tube.  This  gun  is  sighted  at  ball  B  by  look- 
ing through  the  brass  tube,  and  as  ball  A  slips  off  the  end  of 
the  brass  tube  it  comes  against  a  very  light  wire  bridge  which 
spans  across  between  two  brass  clips  thus  breaking  an  electric 
circuit.  This  circuit  includes  an  electromagnet  which  supports 
ball  B. 


Fig.  5. 


CENTER  OF  MASS. 

The  study  of  dynamics  always  begins  with  the  study  of 
'translatory  motion.  Thus  the  discussion  on  pages  3-13,  General 
Physics,  refers  to  translatory  motion.  A  material  particle  is  an 
ideal  body  so  small  that  the  only  sensible  motion  of  which  it  is 
capable  is  translatory  motion,  and  the  term  material  particle  is 
used  in  dynamics  merely  to  direct  one's  attention  narrowly  to 
translatory  motion.  Thus,  if  one  is  concerned  only  with  trans- 
latory motion,  one  may  think  of  a  body  of  any  size  and  shape 
as  a  particle  located  at  the  center  of  mass  of  the  body.  The 
center  of  mass  of  a  body  is  the  point  at  which  a  single  force  must 
be  applied  (that  is  the  line  of  action  of  the  single  force  must  pass 
through  the  center  of  mass)  to  produce  translatory  motion  only. 

9.  Experiment  with  a  slim  stick. — The  experiment  described 
in  Art.  9,  page  14,  General  Physics,  is  quite  effective  if  tried 
before  a  class,  but  every  student  should  try  it  for  himself. 

10.  Experiment  with  hammer  and  stick  as  described  on  page 
15,  General  Physics,  is  extremely  important  as  elucidating,  the 
definition  of  center  of  mass. 

11.  Experiment  showing  the  identity  of  center  of  mass  and 
center  of  gravity. — The  center  of  gravity  of  a  body  is  the  point 
of  application  of  the  single  force  which  is  the  resultant*  of  all 
the  forces  with  which  the  earth  pulls  on  the  various  parts  of  the 
body.     When  a  body  is  supported  by  a  single  force  the  line  of 
action  of  the  force  must  pass  through  the  center  of  gravity  of 
the  bcdy.     Therefore  the  center  of  gravity  of  a  slim  stick  can 
be  located  by  balancing  the  stick  on  a  knife  edge;  and  the  point 
so  found  coincides  with  the  center  of  mass  as  located  in  experi- 
ment 9. 

*  See  Art.   23   of  Franklin  and   MacNutt's  Elementary  Statics,  published  by 
Franklin,  MacNutt  and  Charles,  South  Bethlehem,  Pa. 

22 


TRANSLATORY   MOTION   IN   A   CIRCLE. 


\ 


Fig.  6. 


It  is  important  to  understand  that  translatory  motion  does  not 
mean,  necessarily,  straight  line  motion.  Thus  Fig.  6  indicates 
a  stick  performing  translatory  motion  in  a  circle,  and,  as  explained 
on  page  14,  General  Physics,  the  entire 
stick  may  be  thought  of  as  concentrated 
at  its  center  of  mass  C. 

I 

When  a  ball  is  twirled  on  a  string  the 
ball  itself  makes  one  revolution  every  time 
it  goes  round  the  circle.  This  is  evident 
when  one  considers  that  one  particular 
part  of  the  ball  always  faces  inwards.  The 
motion  of  the  ball  is  therefore  a  combina- 
tion of  translatory  motion  and  rotatory 
motion,  but  the  rotatory  motion  once  established  is  constant 
(unchanging  in  value  and  about  an  axis  which  does  not  change 
its  direction)  whereas  the  translatory  motion  is  variable  (chang- 
ing in  direction). 

12.  Experiment  with  ball  and  string. — The  authors  have  al- 
ways found  it  worth  while  to  go  through  with  the  discussion  of 
Art.  31,  pages  38—40,  General  Physics,  in  the  class  room  with 
ball  and  string  in  hand.  It  is  well  to  point  out  that  the  pull  of 
the  string  is  the  only  force  acting  on  the  ball,  drag  of  air  and  pull 
of  gravity  being  assumed  to  be  negligible. 

A  ball  is  tossed  through  the  air;  what  forces  act  on  the  ball 
after  it  leaves  the  hand?  If  any  one  in  answer  to  this  question 
should  seem  to  think  that  the  continued  motion  of  the  ball 
indicates  a  sort  of  continued  driving  force  exerted  by  the  hand, 
let  him  remember  that  force  means  always  an  actual  push  or  pull 
BY  something  ON  something;  and  very  certainly  the  hand  exerts 
neither  a  push  nor  a  pull  on  the  ball  after  the  ball  has  left  the 

23 


24  CALENDAR  OF  LEADING  EXPERIMENTS. 

hand!  The  gravity  pull  of  the  earth  continues  to  act  on  the 
ball  and  the  air  continues  to  exert  a  backward  drag  on  the  moving 
ball;  and  these  continued  forces  modify  the  motion  of  the  ball 
greatly. 

But  does  not  something  pull  outwards  on  a  ball  which  is  being 
twirled  on  a  string?  If  such  a  force  exists  it  must  be  exerted  by 
something.  The  pull  of  gravity  is  exerted  by  the  earth  (although 
the  connecting  mechanism  between  the  earth  and  the  ball  is 
invisible),  the  backward  drag  of  the  air  on  the  moving  ball  is 
due  to  actual  contact  of  the  air  with  the  ball,  and  the  pull  of  the 
string  on  the  ball  is  evident  to  anybody.  There  is  no  other  force 
acting  on  the  ball. 

Important  example  of  motion  in  a  circle.  The  belt  wheel.— 
A  wheel  is  driven  by  belt  as  shown  in  Fig.  7.  When  the  belt  is 

driven  at  higher  and  higher  speeds, 
it  presses  more  and  more  lightly 
against  the  face  of  the  wheel  and 
slips  more  and  more  easily.  At  a 
certain  critical  speed  the  belt  does 
not  press  against  the  face  of  the 
wheel  at  all. 

Consider  a  short  portion    ab    of 
Fig.  7. 

the  belt.     This  portion  or  particle 

travels  in  a  circular  path  as  it  passes  around  the  pulley.  The 
two  forces  TT  act  on  the  small  portion  due  to  the  tension 
T  of  the  belt,  the  resultant  of  the  two  forces  TT  is  a  force  F 
acting  towards  the  center  of  the  wheel,  and  the  force  F  is  equal 
to  A/  •  T/r,  where  A/  is  the  length  of  the  arc  ab  and  r  is  the 
radius  of  the  wheel.* 

At  zero  belt-velocity,  the  entire  force  F  is  effective  in  pushing 
the  portion  ab  of  the  belt  against  the  pulley  face.  As  the 
velocity  of  the  belt  increases,  a  larger  and  larger  portion  of  F 
is  used  to  constrain  the  portion  ab  to  its  circular  path  or  orbit 

*  See  Franklin  and  MacNutt's  Mechanics  and  Heat,  page  98- 


MECHANICS.  25 

(that  is,  to  produce  the  necessary  acceleration  of  ab  towards 
the  center  of  the  pulley),  and  a  smaller  and  smaller  portion  of 
F  is  available  for  pushing  the  portion  ab  against  the  pulley  face. 
Let  m-Al  be  the  mass  of  the  portion  ab  of  the  belt,  and 
let  v  be  the  velocity  of  the  belt.  Then  the  inward  acceleration 
of  the  portion  ab  is  vz/r  and  the  force  required  to  produce  this 
inward  acceleration  is  m  •  A/  X  v2/r.  Therefore  when 


or  when 


. 

m 


then  the  portion  ab  of  the  belt  is  not  pushed  against  the  pulley 
at  all.  It  is  interesting  to  note  that  this  critical  belt-velocity 
(^T/m)  is  the  velocity  at  which  a  wave  or  bend  would  travel, 
with  unchanged  shape,  along  the  belt.  See  pages  509-511, 
General  Physics. 


MOTION  OF  THE  CENTER  OF  MASS  OF  A  SYSTEM. 

The  idea  of  translatory  motion  can  be  made  to  serve  as  a  basis 
for  the  description  of  any  complicated  motion  whatever;  all 
that  is  necessary  is  to  look  upon  the  moving  body  or  bodies  as 
a  collection  of  particles  and  consider  the  varying  position,  velocity 
and  acceleration  of  each  particle. 

A  collection  of  particles  treated  in  this  way  is  called  a  system 
of  particles  or  simply  a  system.  Thus  a  rotating  wheel  is  a  system 
of  particles,  and  a  portion  of  flowing  water  is  a  system  of  particles. 

When  the  vector  sum  of  all  the  forces  which  act  on  the  particles 
of  a  system  is  zero,  the  center  of  mass  of  the  system  either  remains 
stationary  or  continues  to  move  with  uniform  velocity  in  a  straight 
line.  This  may  be  illustrated  by  considering  the  rotation  of  a 
balanced  emery  wheel.  To  say  that  a  rotating  wheel  is  balanced 
means  that  the  center  of  mass  of  the  wheel  lies  in  the  axis  of  the 
shaft  so  that  the  center  of  mass  remains  stationary  as  the  wheel 
rotates,  and  no  force  need  be  exerted  on  the  shaft  by  the  bearings 
except  the  upward  force  required  to  support  the  wheel  and  axle 
against  the  pull  of  gravity. 

When  the  vector  sum  of  all  the  forces  which  act  on  the  particles 
of  the  system  is  not  equal  to  zero,  then  the  center  of  mass  of  the  system 
is  accelerated  in  the  direction  of  the  resultant  of  all  the  forces,  the 
acceleration  is  proportional  to  the  resultant  of  all  the  forces  and 
inversely  proportional  to  the  total  mass  of  the  system.  Indeed  in 
this  case  we  have  the  equation  F  =  MA,  where  A  is  the 
acceleration  of  the  center  of  mass  of  the  system  of  particles,  F 
is  the  vector  sum  of  all  the  forces  which  act  on  the  system,  and 
M  is  the  total  mass  of  the  system.*  This  may  be  illustrated  by 
considering  the  rotation  of  an  unbalanced  emery  wheel,  an 
emery  wheel  of  which  the  center  of  mass  lies  at  a  distance  r 
from  the  axis  of  rotation.  Then,  as  the  wheel  rotates,  the 

*  See  Franklin  and  MacNutt's  Mechanics  and  Heat,  page  112 

26 


MECHANICS.  27 

center  of  mass  describes  a  circular  path  of  radius  r,  the  accelera- 
tion of  the  center  of  mass  is  equal  to  v2/r  or  to  4?r2w2r  at  each 
instant,  and  a  side  force  equal  to  ^.irVrM  and  parallel  to  r  at 
each  instant  must  act  on  the  axle  to  constrain  the  center  of  mass 
to  its  circular  path,  precisely  as  if  the  entire  mass  of  the  wheel 
were  concentrated  at  its  center  of  mass.  The  force  here  described 
is  the  force  which  must  be  exerted  on  the  emery  wheel  shaft  by 
the  bearings,  and,  of  course,  the  shaft  must  exert  an  equal  and 
opposite  force  on  the  bearings.  This  reacting  force  is  usually 
sufficiently  large  in  the  case  of  a  rapidly  rotating  unbalanced 
wheel  to  cause  very  violent  vibrations  of  the  supporting  structure. 

13.  Emery-wheel   experiment. — Mount   an    iron   disk   on   a 
small  spindle  mounted  in  bearings  and  balance  the  disk  with  a 
removable  bolt  or  cap  screw  near  one  edge  so  that  the  disk  can 
be  easily  unbalanced  by  removing  this  cap  screw.     Remove  the 
spindle  from  its  bearings  and  place  it  upon  two  parallel  level 
straight  edges,  and  show  that  it  is  balanced  whe"n  the  cap  screw 
is  in  place  and  unbalanced  when  the  cap  screw  is  removed.     Then 
replace  the  spindle  in  the  bearings  and  drive  in  balanced  and 
unbalanced  conditions. 

14.  Earth  and  moon  experiment. — The  earth  and  moon  rotate 
about  their  common  center  of  mass  once  every  lunar  month, 
and  this  center  of  mass  describes  a  smooth  elliptical  orbit  about 
the  sun  once  a  year.     This  motion  of  the  earth  and  moon  may 
be  illustrated  as  follows:   A  large  ball  and  a  small  ball  are  tied 
together  with  a  string,  and  the  center  of  mass  of  the  two  balls 
(with  the  string  stretched)  is  marked  by  a  piece  of  red  flannel 
tied  to  the  string.     The  balls  are  then  tossed  through  the  air  in 
such  a  manner  as  to  cause  them  to  rotate  and  keep  the  connecting 
string  stretched  tight,  and  the  piece  of  red  flannel  describes  a 
smooth  curve.     This  experiment  is  not  very  striking  because  the 
eye  naturally,  and  in  spite  of  every  effort  to  the  contrary,  takes 
the  large  ball  as  a  basis  of  reference,  so  that  the  small  ball  and 
the  red  flannel  both  seem  to  rotate  around  the  large  ball. 


SPIN-INERTIA.* 

The  spin-inertia  K  of  a  body  about  a  given  axis  may  be 
defined  as  T/a,  where  T  is  the  torque  (unbalanced)  which 
acts  on  the  body  and  a  is  the  spin-acceleration  produced  by  T 
(the  torque  T  is  of  course  exerted  about  the  given  axis),  or  K 
is  numerically  equal  to  the  torque  T  required  to  produce  unit 
spin-acceleration  (one  radian  per  second  per  second). 

15.  Experiment  with  a  slim  stick. — A  round  stick  about  two 
feet  long  and  half  an  inch  in  diameter  is  tapered  to  a  point  at 
one  end  so  that  it  may  be  set  spinning  by  thumb  and  finger, 
(a)  Show  that  a  moderately  small  torque  acting  for  a  very  short 
time  sets  the  stick  spinning  rapidly  about  its  longitudinal  axis 
(00,     Fig.  15,  General  Physics)  so  that  the  spin-inertia  of  the 
stick  about  the  axis   00  is  small,     (b)  Show  that  a  much  larger 
torque  acting  for  a  longer  time  produces  a  much  slower  spin 
about  a  transverse  axis    (00,    Fig.  16,  General  Physics)  so  that 
the  spin-inertia  of  the  stick  about  the  transverse  axis  is  large. 

1 6.  Direct  determination  of  spin-inertia. — A  large  wheel  is 
mounted  on  a  shaft  of  which  the  radius  is   r   feet,  and  the  shaft 
is  supported  in  ball  bearings.     A  string  is  wound  around  the 
shaft,  a  spring  scale  is  attached  to  the  string,  a  steady  pull  of 
F  poundals  is  exerted  on  the  string  for,  say,  exactly  10  seconds 
(=  /),    and  the  length    /    of  string  unrolled  during  this  time  is 
measured. 

The  torque  T  exerted  on  the  wheel  is  equal  to  Fr,  where  r 
is  the  distance  from  axis  of  shaft  to  middle  of  string,  and  the  spin- 
acceleration  of  the  wheel  is  equal  to  T/K  radians  per  second 
per  second  according  to  equation  (7),  page  22,  General  Physics. 
Therefore  the  spin-velocity  5  which  is  gained  by  the  wheel 
during  /  seconds  is  s  =  (T/K)t.  Now  the  average  spin- velocity 

*  Usually  called  moment  of  inertia. 

28 


MECHANICS.  29 

during  the  t  seconds  is  %s  (see  page  23,  General  Physics)  and 
therefore  the  total  number  of  radians  turned  during  the  time  t 
is  %st  or  %(T/K)t2,  or  the  total  number  of  revolutions  turned 
in  the  time  t  is  %(T/K)P  -T-  2?r  so  that  the  length,  /,  of  string 
unrolled  during  the  time  t  is  /  =  [%(T/K)P  +  2ir}  X  2irr  so 

TPr 
that     /  =  \~~JF~     in  which     K    is  the  only  unknown  quantity 

and  its  value  may  therefore  be  calculated.  The  result  will  be 
expressed  in  pound  (feet).2 

Note  i. — The  force  F  may  be  expressed  in  "pounds,"  the 
torque  T  in  "pound "-feet  and  the  moment  of  inertia  in  slug- 
(feet)2.  See  pages  50  and  51  of  this  volume. 

Note  2. — This  experiment  may  be  arranged  in  accordance 
with  the  discussion  in  Art.  34,  General  Physics. 

Note  3. — The  wheel  above  mentioned  may  be  a  circular  disk 
of  solid  iron  so  that  its  spin-inertia  may  be  calculated  by  the 
formula  given  in  the  table  on  page  22,  General  Physics  (the 
spin-inertia  of  the  shaft  may  be  neglected  because  the  above 
measurement  is  not  very  precise)  and  thus  the  experimentally 
determined  value  of  K  may  be  compared  with  its  calculated 
value. 

Note  4. — A  very  instructive  modification  of  this  experiment 
is  as  follows:  Instead  of  the  wheel  let  four  slim  rods  be  fixed  to 
the  shaft  like  the  spokes  of  a  wheel,  and  determine  the  value 
of  K  of  this  structure  as  explained  above.  Then  attach  massive 
steel  balls  to  the  spoke-like  arms  each  at  a  distance  p  feet  from 
the  axis  of  the  shaft,  and  determine  anew  the  spin-inertia  K'. 
Then  Kf  —  K  will  be  found  equal  to  mp2  where  m  is  the 
combined  mass  of  the  steel  balls.  It  is  best,  however,  according 
to  the  authors'  experience,  relegate  quantitative  experiments 
to  the  laboratory  where  the  student  makes  all  the  measurements. 

Derivation  of  the  equation  K  =  Sr2-Aw. — The  student  of 
mathematics  is  usually  led  to  interpret  the  equation 


30  CALENDAR  OF  LEADING   EXPERIMENTS. 

fofi-dx  =  J*3  +  C 
thus 

f  The  function  whose  deriva-  1 

i  f  ==    x    I   C~, 

I  tive  with  respect  to  x  is  x2  \ 

and  from  this  point  of  view  it  is  easy  to  determine  the  spin- 
inertia  K  of  a  body  of  regular  shape  as  on  pages  594-597, 
General  Physics,  where  the  derivative  of  K  with  respect  to  a 
chosen  variable  is  derived,  etc.  It  is  easy  to  alter  this  discussion 
so  as  to  use  equation  (7),  page  (22),  General  Physics,  namely, 
T  =  Ka,  instead  of  the  equation  W  =  %Ks2.  Consider  the 
added  mass  2irAr'Ar\  its  sidewise  velocity  (at  right  angles 

to    r)    is    rs,    its  sidewise  acceleration  is    r  -r    or    ra,    and  the 

sidewise  force  AF  which  must  act  on  it  to  accelerate  it  is 
AF  —  2-jrAr-Ar  X  ra.  Therefore  a  portion  AT  of  the  total 
torque  T  (=  Ka)  which  is  acting  on  the  body  is  used  to 
accelerate  the  added  material,  namely, 

AT  =  r-AF  =  2irAr-Ar  X  ra  X  r. 

Therefore  since  AK  =  AT /a  we  have  AK  =  2wAr3-Ar  so  that 
the  derivative  of  K  with  respect  to  r  is  2irAr3. 

The  more  general  equation  K  =  Sr2-Aw  may  be  derived  as 
follows:  Let  us  consider  a  small  particle  of  the  body  of  mass 
Am  at  a  distance  r  from  the  axis  of  rotation.  The  sidewise 
velocity  of  Am  (at  right  angles  to  r)  is  rs  and  the  sidewise 
acceleration  of  Am  is  ra.  Therefore  the  sidewise  force  which 
must  act  on  Am  is  AF  =  Am  X  ra  and  the  torque  value  of  this 
force  about  the  axis  of  rotation  is  AT  =  Am  X  ra  X  r,  so  that 
the  total  torque  T  acting  to  produce  spin-acceleration  is 
T  =  a2r*-Am  so  that  K  =  2>2-Aw. 


THE  GYROSCOPE  OR  GYROSTAT. 


It  is  scarcely  possible  to  bring  out  the  simple  meaning  of  the 
discussion  of  Figs.  390,  396  and  39^,  General  Physics  without 
showing  the  experiment  as  follows: 

17.  Experiment  with  a  bicycle  wheel. — (a)  Hold  a  bicycle 
wheel  (not  spinning)  as  indicated  in  Fig.  8,  supporting  both  ends 


west     O 


north 


north 


west 


Fig.  8. 

of  the  spindle  by  the  hands.  Release  the  spindle  at  one  end 
and  keep  it  supported  at  end  0,  catch  it  again  immediately 
and  call  attention  to  the  fact  that  eastward  velocity  has  been 
given  to  every  particle  in  the  upper  half  of  the  wheel  and  westward 
velocity  to  every  particle  in  the  lower  half  of  the  wheel  by  the  unbal- 
anced torque  due  to  the  gravity  pull  of  the  earth  on  the  wheel. 

(b)  Hold  the  spinning  wheel  and  release  it  in  the  same  way, 
catch  it  again  immediately  and  call  attention  to  the  fact  that 
eastward  velocity  has  been  given  to  every  particle  in  the  upper  half 
of  the  wheel  and  westward  velocity  to  every  particle  in  the  lower 
half  of  the  wheel  by  the  unbalanced  torque  due  to  the  gravity  pull  of 
the  earth  on  the  wheel. 

(c)  Hold  the  spinning  wheel  and  release  it  as  before  and  allow 


32  CALENDAR  OF  LEADING  EXPERIMENTS. 

it  to  precess  steadily,  and  point  out  the  meaning  of  Fig.  38, 
General  Physics. 

The  essential  steps  in  the  analysis  of  the  sidewise  accelerations 
of  the  particles  of  a  rotating  disk  in  precession  are  given  on  pages 
168-169  of  Franklin  and  MacNutt's  Mechanics  and  Heat. 

There  are  two  distinct  aspects  of  the  problem  of  the  simple 
gyroscope,  namely,  (a)  To  determine  the  torque  required  to 
produce  a  specified  precession,  and  (b)  To  determine  the  pre- 
cession produced  by  a  given  torque.  The  first  aspect  of  the 
problem  is  very  much  simpler  than  the  second.  Indeed  the 
first  aspect  is  a  problem  in  differentiation,  that  is  to  say,  it  turns 
out  to  be  such  when  it  is  fully  developed;  whereas  the  second 
aspect  turns  out  to  be  a  problem  in  integration.  Now  any  prob- 
lem in  integration  involves  the  evaluation  of  integration  constants 
(on  the  basis  of  initial  conditions  supposed  to  be  given),  and  the 
oscillatory  movements  of  the  gyroscope  as  very  briefly  described 
on  pages  42  and  43  of  this  volume  are  related  to  these  integration 
constants. 

Why  is  it  that  one  does  not  use  the  moment  of  inertia  about 
the  axis  T  in  Fig.  37,  General  Physics,  in  the  equation  T  —  Ka? 
One  certainly  would  use  the  moment  of  inertia  about  T  if  the 
wheel  were  not  already  spinning!  Answer:  One  does  use  the 
moment  of  inertia  about  T  and  several  things  not  mentioned 
in  the  bob-tailed  theory  (the  simple  theory  as  given  in  General 
Physics'),  unless  one  thinks  of  problem  (b)  as  the  simple  inverse 
of  problem  (a). 


WHAT  IS  LACKING  IN  THE  DISCUSSION  OF  FIGS.  140, 
AND    146   IN   General  Physics* 

It  is  assumed  in  the  discussion  of  Figs.  140,  and  146  and  again 
in  the  discussion  of  Figs.  37  and  38  in  General  Physics  that  spin- 
velocities  can  be  added  by  the  parallelogram  law.  The  proof  of 
this  is  outlined  in  Franklin  and  MacNutt's  Mechanics  and  Heat, 
page  176. 

It  is  the  validity  of  the  parallelogram  law  that  justifies  the 
representation  of  spin-velocity  by  a  line  or  arrow  as  specified 
on  page  19,  General  Physics.  Similarly  the  parallelogram  law 
must  be  valid  for  the  addition  of  torques  to  justify  the  repre- 
sentation of  a  torque  by  a  line  or  arrow  as  specified  on  page  19 
of  General  Physics.  The  demonstration  of  the  parallelogram 
law  for  the  addition  of  torques  is  given  in  Franklin  and  Mac- 
Nutt's Mechanics  and  Heat,  page  177. 

It  is  utterly  inadequate  to  say  that  spin-velocity  and  torque 
are  "directed  quantities  and  therefore  to  be  added  by  the 
parallelogram  law."f  This  is  evident  when  we  consider  that  a 
displacement  of  a  body  through  a  certain  angle  <j>  about  a 
given  axis  is  not  to  be  distinguished  in  its  mode  of  specification 
from  such  quantities  as  spin-velocity  and  torque,  whereas  angular 
displacements  cannot  be  added  by  the  parallelogram  law. 

*  One  thing  that  is  lacking  grows  out  of  the  assumption  that  problem  (6)  as 
described  on  page  32  is  the  simple  inverse  of  problem  (a).  This  deficiency  most 
beginners  will  accept  uncomplainingly,  we  are  sure  of  that  because  we  have  men- 
tioned integration  in  connection  therewith.  We  refer  here,  in  particular,  to 
deficiencies  that  are  comparatively  easy  to  supply,  but  which  simply  must  not  be 
lost  sight  of  by  accepting  plausibility  for  rigor. 

f  Professor  Rettger's  review  of  the  second  edition  of  Dadourian's  Analaytical 
Mechanics  (Science,  August  26,  1916,  page  279)  calls  attention  to  a  fault  which  be- 
comes more  serious  in  Professor  Dadourian's  later  attempt  at  defense.  Plausi- 
bility is  in  many  cases  necessary  in  the  presentation  of  a  mathematical  subject  to 
young  men,  but  to  think  of  plausibility  as  the  same  thing  as  rigor  is  fatal. 

4  33 


34  CALENDAR  OF  LEADING  EXPERIMENTS. 

One  of  the  best  discussions  of  vector  quantity  is  to  be  found  in 
the  first  chapter  of  Abraham  and  Foppl's  Theorie  der  Electrizitdt, 
Vol.  I,  third  edition,  Leipzig,  1907. 

Science,  even  in  its  elements,  presents  serious  difficulties. 
Anyone  who  thinks  otherwise  knows  only  pseudo-science  as 
P.  G.  Tait  has  said,*  and  pseudo-science  is  the  bane  of  science 
teaching.f  Indeed  the  most  serious  fault  with  our  physics  teach- 
ing is  that  our  physicists  who  know  all  that  has  been  thought 
and  all  that  has  been  done  in  physics  (and  there  are  many  of  our 
physicists  who  do)  hold  themselves  aloof  while  half-educated 
men,  as  in  the  recent  rehash  of  elementary  mechanics  in  Science, 
talking  as  oracles  from  out  of  their  dignified  institutional  settings, 
make  pi  of  everything! 

*  See  introduction  to  Tait's  Heat. 

t  The  most  convincing  statement  of  this  fact  has  been  made  by  a  student  of 
the  Classics.  See  remarkable  article  by  Paul  Shorey  in  the  Atlantic  Monthly 
for  June  and  July,  191?. 


THE  USE  OF  THE  GYROSCOPE  AS  A  RUDDER 
CONTROL. 

Consider  a  freely  precessing  gyroscope,  the  precession  being  due 
to  the  pull  of  gravity.  If  the  precession  is  hindered  by  letting  the 
end  of  the  gyroscope  axle  (see  Fig.  39^,  General  Physics}  come 
against  a  stop,  the  gyroscope  drops  at  once  like  any  ordinary  inert 
body.  In  order  that  the  gyroscope  may  stand  out  horizontally 
in  spite  of  the  downward  pull  of  gravity,  the  gyroscope  must  be 
free  to  precess  about  a  vertical  axis. 

An  important  use  of  the  gyroscope  is  for  automatically  con- 
trolling the  rudder  of  a  torpedo  or  of  a  flying  machine.  In 
such  a  case  the  gyroscope  constitutes  a  frame-work  which  keeps 
an  unchanged  direction  in  space,  and  a  sudden  veering  of  the 
torpedo  or  flying  machine  to  right  or  left  brings  the  gyroscope 
frame  against  a  lever  which  opens  a  valve  and  admits  compressed 
air  to  a  piston  which  shifts  a  rudder.  To  exert  a  force  in  opening 
the  valve  the  gyroscope  must  be  entirely  free  to  precess  in  a 
plane  at  right  angles  to  the  direction  of  the  force,  exactly  as  in 
the  case  of  a  gyroscope  precessing  under 
the  pull  of  gravity  as  above  explained. 

Experiment  18. — Support  one  end  of  the 
axle  of  a  spinning  bicycle  wheel  in  the  hand, 
allow  the  other  end  of  the  axle  to  come 
against  the  side  of  a  vertical  post,  and  call 
attention  to  the  fact  that  the  axle  drops 
exactly  as  if  the  wheel  were  not  spinning, 
or  would  so  drop,  if  the  powerful  reaction 

Fig.  11. 

did  not  move  the   hand-support  sidewise 

and  if  it  were  not  for  the  great  friction  due  to  the  very  large  force 

with  which  the  moving  end  of  the  axle  is  pressed  against  the  post. 

19.  Curious  gyroscope  toy. — When  the  moving  end  of  the 
axle  of  a  precessing  gyroscope  comes  against  an  obstacle  (even 

35 


36  CALENDAR  OF  LEADING  EXPERIMENTS. 

when  no  outside  torque  is  acting  to  push  the  axle  against  the 
obstacle)  the  axle,  if  it  is  rotating  with  the  wheel,  starts  to  roll 
along  the  obstacle.  This  motion  (precession)  of  the  axle  causes 
a  reaction  which  pushes  the  axle  against  the  obstacle  so  that 
the  rolling  axle  will  follow  an  obstacle  of  almost  any  shape. 
Fig.  II  shows  a  very  interesting  toy  in  which  the  axle  of  a 
spinning  top  rolls  round  and  round  the  entire  boundary  of  a 
double  spiral  made  of  a  strip  of  metal. 


WHAT  MAKES  A  HARD-BOILED  EGG  RISE  ON  END 
WHEN   SPUN   ON   ITS  SIDE? 

Experiment  20. — Spin  a  sharp-pointed  top  with  its  axle  in- 
clined and  note  its  motion  of  precession.  This  precession  is 
caused  by  the  tipping  action  or  torque  action  produced  by 
the  pull  of  gravity,  and  the  precession  produces  a  reaction  which 
is  just  sufficient  to  keep  the  axis  of  the  top  at  a  constant  incli- 
nation. 

Increase  the  speed  of  the  precessional  motion  by  holding  a 
thin  rod  above  the  top  and  helping  the  precession.  This  in- 
creased precession  produces  more  than  enough  reaction  to  support 
the  inclined  top  against  gravity,  and  therefore  the  axis  of  the  top 
rises  to  the  vertical. 

The  rolling  of  the  blunt  end  of  a  spinning  top  (with  its  axis 
inclined)  on  the  table  hastens  the  precessional  motion  of  the 
top  by  producing  a  side  force  against  the  lower  end  of  the  axle  of 
spin,  and  this  hastening  of  the  precession  soon  brings  the  axis 
of  the  top  into  a  vertical  position. 

The  above  effects  can  be  easily  shown  by  spinning  two  tops 
which  are  alike  except  that  one  has  a  very  blunt  point  and  the 
other  a  sharp  point.  A  hard-boiled  egg  spun  on  its  side  quickly 
stands  up  on  one  end,  the  action  being  the  same  as  in  case  of  the 
blunt  pointed  top. 


37 


THE  SCOPE   OF  DYNAMICS. 

It  may  seem  that  an  excessive  amount  of  space  is  devoted  to 
gyroscopic  motion  in  General  Physics  and  in  this  Calendar,  but 
rational  mechanics  is  an  important  subject  and  it  must  be 
adequately  treated !  The  plan  which  is  commonly  followed  in  our 
technical  schools  is  to  give  a  course  in  analytical  mechanics  to 
supplement  an  altogether  inadequate  course  in  elementary 
mechanics;  and  this  plan  is  worse  than  useless  for  two  reasons, 
namely,  (a)  Because  analytical  mechanics,  the  oldest  branch  of 
theoretical  physics,  has  less  to  do  with  engineering  and  less  to 
do  with  research  than  any  other  branch  of  mathematical  physics, 
and  (b)  Because  the  limitations  of  the  engineering  student  and 
the  limitations  of  the  engineering  teacher  have  together  led  to 
the  development  of  a  course  in  analytical  mechanics  which  it  is 
a  kindness  to  call  mere  formal  nonsense.  It  sickens  the  brilliant 
student  and  stupefies  those  who  are  not  brilliant;  and  a  close 
competitor  is  the  accepted  course  in  thermodynamics  which 
has  been  developed  under  the  same  hopeless  limitations;  but, 
according  to  our  experience,  there  is  no  group  of  teachers  so 
stiffly  proud  of  what  they  do  as  the  teachers  of  theoretical 
mechanics  and  thermodynamics  in  our  technical  schools. 

What  are  authors  to  do  who  arrange  an  elementary  treatise 
on  dynamics?  Translatory  motion  must  be  taken  up  first, 
then  rotatory  motion,  then  oscillatory  and  wave  motion,  and 
then,  were  it  not  for  the  limitations  of  the  student,  a  touch  of 
statistical  mechanics  which  sticks  its  nose  into  Everything! 
But  we  know  the  imperturbable  satisfaction  of  Deans  of  Engineer- 
ing; what  are  we  to  do?  Having  come  to  detest  the  technical 
school — no,  not  the  school  but  the  elaborate  parts  of  its  cur- 
riculum, we  cannot  be  satisfied  merely  to  expound  the  elementary 
theory  of  dynamics  (we  are  willing  to  stop  short  of  statistical 
mechanics)  and  contemplate  mortality  tables! 

38 


MECHANICS.  39 

The  old  conflict  in  education  was  between  the  older  classics 
and  science,  but  the  conflict  is  now  between  science  and  pseudo- 
science.  We  have  the  watery  pseudo-science  of  the  high  school 
and  college  which  is  intended  to  be  enticing  and  plausible;  and 
we  have  the  pseudo-science  of  the  technical  school  which  is 
exacting  but  unintelligible,  Dryer  than  Dust.  It  would  desiccate 
the  Ocean  if  it  were  not  far  Up  in  the  Air. 


THE  BRENNAN  MONO-RAIL  CAR. 


A  gyrostat  wheel  is  mounted  on  a  horizontal  spindle,  this 
spindle  is  carried  in  a  frame  FF,  Fig.  12,  this  frame  is  free  to 
turn  about  a  vertical  axis,  being  pivoted  in  an  outer  frame  GG 
which  is  fixed  to  a  platform  PP,  and  this  platform  stands  on 
two  pointed  legs  0  (one  leg  is  behind  the  other  in  the  figure). 

The  platform  PP  with  attached 
frame  GG  is  shown  as  slightly 
inclined  to  the  left  in  Fig.  12  so  that 
the  pull  of  the  earth  on  the  entire 
arrangement  exerts  upon  it  a  torque 
Tj  and  the  effect  of  this  torque  is  to 
make  the  spinning  wheel  precess, 
bringing  the  end  a  of  the  spindle 
towards  the  reader.  The  platform 
does  not  fall  over  for  a  compara- 
.  tively  long  time  (while  the  preces- 
sion is  taking  place  as  stated) ,  and 
the  essential  feature  of  the  Brennan 

balancer  is  a  device  for  automatically  hastening  this  precession. 
The  reaction  of  the  unhastened  precession  is  just  sufficient 
to  balance  the  torque  T,  but  the  reaction  of  the  hastened 
precession  is  more  than  enough  to  balance  T,  and  there- 
fore the  hastened  precession  lifts  the  end  b  of  the  spindle. 
Indeed  in  the  Brennan  balancer  the  lifting  of  the  end  b  of  the 
spindle  is  always  enough  to  reverse  the  tilt  of  the  platform  and 
reverse  the  torque  T,  whereupon  the  precession  is  reversed. 
This  reversed  precession  is  allowed  to  continue  until  the  end 
a  of  the  spindle  has  come  back  into  the  plane  of  the  paper,  and 
then  the  reversed  precession  is  hastened,  the  platform  is  tilted 

40 


O 

Fig.  12. 


MECHANICS.  41 

back  to  the  left  (as  shown  in  the  figure),  the  end  a  of  the  spindle 
comes  forwards  to  the  plane  of  the  paper,  the  precession  is  then 
again  hastened,  and  so  on  over  and  over  again.  A  number  of 
devices  have  been  used  for  automatically  hastening  the  precession 
as  stated. 

The  only  advantage  of  the  mono-rail  is  that  a  car  (if  it  is  kept 
erect)  would  run  with  less  vibration  on  one  crooked  rail  than  on 
two  crooked  rails!  It  certainly  is  better,  however,  to  build  two 
straight  rails  and  use  ordinary  cars  than  it  would  be  to  save  a 
little  on  a  crooked  mono-rail  and  spend  a  lot  on  complicated  and 
more  or  less  unreliable  balancing  mechanisms!  In  fact  the  great 
problem  in  heavy-traffic  railroading  is  the  problem  of  making 
the  track  strong  enough  to  carry  the  enormously  heavy  engines 
and  trains,  and  mono-rail  construction  would  be  an  engineering 
absurdity. 

The  gyro-platform. — Two  Brennan  balancers  can  be  arranged 
to  keep  a  platform  very  nearly  level  on  board  ship,  and  such  a 
gyro-platform  may  come  to  have  important  uses. 

Experiment  21. — A  person  sitting  on  the  platform  PP  in 
Fig.  12  can  maintain  his  balance  easily  by  properly  hastening 
the  precession  of  the  heavy  lead-rimmed  bicycle  wheel  which 
has  been  set  spinning  by  hand,  especially  if  the  two  platform 
legs  are  short. 

22.  The  Schlick  device  for  the  steadying  of  a  ship  at  sea. 
Figure  13  shows  a  demonstration  model  of  this  device.  The 
ring  swings  freely  about  a  horizontal  precession  axis  pq,  and  the 
axis  of  spin  of  the  wheel  is  at  right  angles  to  pq  as  shown. 
The  boat  is  represented  by  the  flat  board  AB  which  stands  on 
two  legs  00. 

Take  the  strings  55  in  the  hands  and  exert  a  repeatedly 
reversed  torque  on  the  model,  tending  to  rock  the  boat  back 
and  forth  about  00  as  an  axis. 

Let  us  suppose  that  the  axis  of  spin  of  the  wheel  is  vertical 


42  CALENDAR  OF  LEADING   EXPERIMENTS. 

at  the  beginning,  then  pulling  on  either  string  does  not*  tip  the 
boat  but  causes  precession  about  the  axis  pq.  A  steady  pull 
on  either  string  soon  brings  the  axis  of  spin  into  a  horizontal 
fore-and-aft  position,  and  then  the  boat  tips  freely.  The 
gryostat  is  only  effective  in  preventing  the  tipping  or  rocking  of 


T 


Fig.  13. 

the  boat  when  repeatedly  reversed  pulls  are  exerted  on  the 
strings,  and  the  intensity  and  duration  of  a  single  pull  must  not 
be  sufficient  to  bring  the  axis  of  spin  into  the  horizontal  fore-and- 
aft  position. 

23.  Gyrostatic  elasticity. — A  spinning  gyroscope  is  held  in  the 
position  shown  in  Fig.  14,  and  we  will  speak  of  the  end  of  the  axle 
which  is  opposite  to  0  as  the  outer  end.  The  support  of  the 
outer  end  is  suddenly  removed.  To  establish  the  precessional 
motion  ft  the  ring  and  wheel  must  of  course  be  set  in  motion 
about  a  vertical  axis,  and  this  means  inertia  reaction  whose  effect, 
while  it  lasts,  is  exactly  equivalent  to  an  outside  torque  opposing 

*  Except  for  an  elastic-like  yielding  which  immediately  recovers  when  the  pull 
See  following  discussion  of  gyrostatic  elasticity. 


MECHANICS.  43 

12.  Therefore  the  outer  end  of  axle  and  ring  drops  slightly  while 
precessional  motion  is  being  established.  By  the  time  12  has 
reached  its  normal  mean  value  (the  value  due  to  gravity  torque 
on  ring  and  wheel)  the  downward  velocity  of  outer  end  is  a 
maximum  and  the  downward  momentum  of  ring  and  wheel  carries 
the  outer  end  still  farther  downwards  and  doubles  the  value  of 
12.  Then  the  unbalanced  reaction  of  this  doubled  12  (the  reaction 
due  to  mean  value  of  12  is  just  sufficient  to  balance  the  gravity 
torque)  lifts  the  outer  end  to  its  initial  level  and  reduces  12  to 
zero  as  at  the  beginning  (the  outer  end  having  moved  forwards, 
however,  in  the  direction  of  12).  The  action  as  described  is 
repeated  over  and  over  again. 


Fig.  14. 

When  one  of  the  strings  in  Fig.  13  is  pulled  suddenly  the  boat 
tips  slightly  while  the  precession  is  being  established,  and  if  the 
pull  then  ceases  the  momentum  which  is  associated  with  the 
established  precession  causes  the  precession  to  persist  for  a  short 
time,  and  the  unbalanced  reaction  of  this  continued  precession 
brings  the  boat  again  to  a  level  position.  If  the  strings  are 
removed  in  Fig.  13  a  sudden  blow  causes  the  boat  to  oscillate 
back  and  forth  on  its  two  legs  exactly  as  if  the  boat  were  held 
in  a  vertical  position  by  an  elastic  spring,  and  this  oscillatory 
motion  of  the  boat  is  accompanied  by  an  oscillatory  precession 
of  the  gyrostat.* 

*  A  very  simple  theoretical  discussion  of  this  kind  of  oscillatory  motion  is  given 
by  W.  S.  Franklin,  Physical  Review,  Vol.  XXXIV,  pages  48-52,  January,  1912. 


CONSTANCY    OF    SPIN-MOMENTUM    OF    A    CLOSED 

SYSTEM. 

When  a  body  or  system  of  bodies  is  not  acted  upon  by  any 
forces  from  outside  the  body  or  system,  the  body  or  system  of 
bodies  constitutes  what  is  called  a  closed  system.  The  only 
force  actions  in  a  closed  system  are  the  mutual  force  actions 
between  the  parts  of  the  system. 

The  total  momentum  of  a  closed  system  cannot  change.  This 
is  evident  when  we  consider  two  things,  namely,  (a)  That  an 
unbalanced  force  acting  on  a  body  causes  the  momentum  mv 
of  the  body  to  change,  and  that  the  rate  of  change  of  the  momen- 

d(mv)  .  dv    , 

turn  — i —  is  equal  to  the  force  m  —    because  m  is  a  constant, 
at  at 

and  (b)  That  any  mutual  force  action  between  two  bodies  in  a 
system  consists  of  a  pair  of  equal  and  opposite  forces  so  that  the 
rates  of  change  of  the  momenta  of  the  two  bodies  must  be  equal 
and  opposite  so  that  their  combined  momentum  does  not  change. 
The  spin-momentum  of  a  body  is  equal  to  Ks,  where  K  is 
the  spin-inertia  of  the  body  and  5  is  its  spin-velocity.  In  a 
closed  system  a  mutual  torque  action  between  two  parts  of  the 
system  means  the  exertion  of  equal  and  opposite  torques  on  the 
two  parts,  and  equal  and  opposite  rates  of  change  of  spin- 
momentum  by  the  two  parts.  Therefore  the  total  spin-momen- 
tum of  a  closed  system  cannot  change. 

24.  Experiment  with  a  pivoted  stool. — A  platform  or  stool  is 
supported  on  a  ball-bearing  pivot.  Stand  on  this  platform  with 
weights  in  the  hands  and  with  arms  outstretched,  and  have  an 
assistant  set  you  rotating  slowly  about  a  vertical  axis.  Then 
draw  in  your  arms  thus  greatly  decreasing  the  spin-inertia  K 
of  the  system.  The  spin-momentum  Ks  remains  constant 
and  the  spin-velocity  s  is  greatly  increased  when  K  is  reduced. 

44 


MECHANICS.  45 

25.  Experiment  with  emptying  bowl. — A  deep  bowl,  preferably 
of  glass,  is  filled  with  water  and  the  water  is  set  slowly  rotating 
by  stirring  the  water  with  a  stick.     A  hole  at  the  center  of  the 
bottom  of  the  bowl  is  then  opened  and  the  water  is  allowed  to 
run  out.     The  water  in  the  bowl  flows  to  some  extent  towards 
the  axis  of  the  bowl  and  the  spin-inertia  of  the  system  (water  in 
the  bowl  and  the  long  "  rod  "  of  water  which  is  to  be  thought  of 
as  coming  out  of  the  hole)  decreases  greatly.     Therefore  the  slow 
rotatory  motion  of  the  water  is  greatly  increased  as  shown  by 
the  production  of  a  deep  whirl  at  the  center  of  the  bowl.     See 
description  of  cyclone  and  tornado,  page  91,  General  Physics. 

26.  Experiment  with  pivoted  stool  and  heavy  rimmed  bicycle 
wheel. — Stand  on  the  pivoted  stool,  hold  the  bicycle  with  its 
axis  horizontal  and  set  it  spinning  rapidly.     Then  when  the  axis 
of  the  spinning  wheel  is  brought  into  a  vertical  position  your  body 
will  be  set  rotating  in  a  direction  opposite  to  the  spinning  wheel. 
Assuming  the  pivot  of  stool  to  be  frictionless  no  torque  about 
a  vertical  axis  can  be  exerted  on  your  body  from  outside,  and 
therefore  the  spin-momentum  of  the  pivoted  system  about  a 
vertical  axis  must  remain  what  it  was  at  first,  namely,  zero. 


STATIC   AND   DYNAMIC   BALANCING. 


A  wheel  and  axle  is  said  to  be  statically  balanced  when  the 
center  of  mass  (or  center  of  gravity)  of  the  whole  lies  on  the 
axis  of  the  axle.  Such  a  wheel  and  axle  will  stand  indifferently 
in  any  position  when  the  axle  rests  on  two  parallel,  straight, 
horizontal  rails. 

A  wheel  and  axle  is  said  to  be  dynamically  balanced  when  the 
axle  has  no  tendency  to  quiver  when  the  whole  is  spinning 
rapidly. 

Figure  15  shows  a  body  B  mounted  on  a  shaft,  and  the 
center  of  mass  C  of  the  whole  is  on  the  axis  of  the  shaft;  that 
is  to  say,  the  body  is  statically  balanced.  But  if  the  shaft 


B 


XC 


shaft 


\rope 


Fig.  15. 


Fig.  16. 


is  supported  in  bearings  and  the  body  set  rotating,  the  bearings 
must  exert  very  considerable  forces  upon  the  shaft,  and  the  arrows 
FF  show  the  directions  of  these  forces  for  the  given  instantaneous 
position  of  the  body;  or  if  hung  by  a  rope  as  shown  in  Fig.  16 
and  set  spinning,  the  shaft  will  take  the  position  shown. 

46 


MECHANICS. 


47 


Dynamic  balancing  of  a  dynamo  armature. —  A 
thin  disk  or  narrow  wheel  and  axle  can  be  brought 
into  dynamic  balance  by  placing  the  axle  on  hori- 
zontal rails  and  adding  material  to  or  taking 
material  from  one  edge  or  the  other  until  the 
wheel  and  axle  is  in  static  balance.  But  a  long 
body  like  a  dynamo  armature,  having  been  static- 
ally balanced,  must  be  hung  as  shown  in  Fig.  17 
and  set  spinning  rapidly  about  a  vertical  axis. 
If  the  armature  is  dynamically  out  of  balance  op- 
posite sides  a  and  b  of  the  shaft  will  be  whit- 
ened by  chalk  held  lightly  against  the  spinning 
shaft.  Equal  weights  must  then  be  added  at  c 
and  d  (or  taken  away  from  e  and  / ) ;  and  this 
operation  is  repeated  until  the  balancing  is  satisfactory. 


rope 


17. 


ONE,   OILER,   KNEW   IT  ALL! 

The  motion  of  a  symmetrical  spinning  body  like  a  wheel  or  a 
top  is  extremely  simple  as  compared  with  the  motion  of  a  non- 
symmetrical  body,  and  any  one  who  cannot  digest  Euler's 
(pronounced  Oiler's)  equations  should  go  to  Poinsot's  wonderful 
paper  entitled  Theorie  Nouvelle  de  la  Rotation  des  Corps,  or  make 
for  himself  one  of  the  non-symmetrical  tops  which  Maxwell 
devised  many  years  ago.  It  is  a  common  remark  among  thorough 
students  of  rigid  dynamics  that  Euler  knew  it  all!  and  he  did,  in 
fact,  know  it  all. 

Anyone  can  know  what  Euler  knew  by  taking  some  pains,  and 
to  one  who  knows  what  Euler  knew  it  is  extremely  funny,  and 
more  than  funny,  to  read  the  following  letter  which  was  addressed 
to  the  editor  of  an  important  scientific  journal  by  a  director  of 
the  Aero  Club  of  America  asking  an  utterly  meaningless  question 
about  the  simplest  aspect  of  rotatory  motion  with  a  pretense  of 
forty  thousand  dollars'  worth  of  seriousness;  for,  in  fact,  the 
Cash  Prize  faded  to  nothing  when  investigation  was  made. 

Dear  Sir:  I  respectfully  state  that  the  following  question  is 
before  the  members  of  the  Aero  Club  of  America  for  solution 
and  I  would  appreciate  it  very  much  if  you  would  kindly  put 

the  question  to  the  readers  of  through  its  valuable 

columns : — 

"What  is  the  acceleration  of  precession  when  mass  spins  and 
precesses  with  the  same  radius  vector,  and  in  the  same  plane, 
tangential  to  the  earth's  surface?" 

The  above  question  is  important  and  is  put  in  consideration 
of  $40,000  Cash  Prize  offered. 

I  remain,  Yours  for  the  advancement  of  science, 

1  Director. 

48 


MECHANICS.  49 

Ours  for  the  advancement  of  science!  Yes,  now,  ours.  He 
means  it  with  all  the  good  will  he  can  muster  without  cost  or 
pains.  But  he  was  at  one  time  yours,  gentle  teacher,  whether  of 
the  technical  school  or  college,  and  he  left  you,  untouched  by 
those  kindly  but  rigorous  exactions,  which  when  shot  through 
and  through  with  sympathetic  intelligence  do  bring  young  men 
to  the  pains  of  those  who  really  learn  and  brace  them  for  the 
grief  of  those  who  are  wise. 


WORK  AND   POWER. 

The  difficulty  of  the  precise  ideas  of  work  and  power  as  applying 
to  actual  conditions  and  things  is  amusingly  illustrated  by  the 
following  problem  which  we  gave  to  a  group  of  engineering 
students  at  the  end  of  a  half-year's  course  in  elementary 
mechanics.  "A  cart  moves  due  northwards  at  a  velocity  of 
5^  feet  per  second.  A  man  pushes  vertically  downwards  on  the 
cart  with  a  force  of  200  pounds,  and  a  mule  pulls  due  northwards 
on  the  cart  with  a  force  of  50  pounds.  Find  the  rate  at  which 
the  man  does  work  and  the  rate  at  which  the  mule  does  work." 
We  deliberately  refrained  from  naming  the  part  of  the  man's 
body  which  exerted  the  force  on  the  cart!  In  answer  to  the 
question  44  per  cent  of  the  young  men  found  that  the  man 
developed  2  horse-power  and  the  mule  developed  J  horse-power. 
Truly,  mule  driving  would  be  strenuous  labor  for  our  pampered 
college  students! 

Another  example  illustrates  the  dislike  of  many  young  men 
for  the  hardest  labor  in  the  world,  namely,  thinking!  "The 
force  required  to  tow  a  boat  is,  let  us  say,  proportional  to  the 
velocity  of  the  boat.  To  tow  the  boat  at  a  velocity  of  2  miles 
per  hour  requires  2  horse-power;  how  much  power  would  be 
required  to  tow  the  boat  at  a  velocity  of  4  miles  per  hour?" 
More  than  95  per  cent  of  the  class  accepted  unthinkingly  the 
suggestion  of  simple  proportion  and  got  4  horse-power  as  the 
answer ! 

Kinetic  energy  and  the  engineer's  system  of  units.  We 
usually  speak  of  the  system  of  units  which  is  based  on  the  foot 
as  the  unit  of  length,  the  earth-pull-on-a-one-pound-body-in- 
London  as  the  unit  of  force  and  the  second  as  the  unit  of  time 
as  the  foot-slug-second  system  (f.s.s.  system)  for  the  sake  of  the 
sharp  differentiation  which  is  thereby  made  between  this  system 

50 


MECHANICS.  51 

and  the  foot-pound-second  system  (f.p.s.  system).  The  units  of 
any  system  may  be  used  in  any  equation  in  mechanics  (including 
hydrostatics  and  hydraulics)  if  the  equation  is  in  its  simple  gener- 
alized form.  Thus  in  the  f.s.s  system  the  kinetic  energy  of  a 
moving  body  in  foot- "pounds"  is  equal  to  ^mv2  where  m  is 
the  mass  of  the  body  in  slugs  and  v  is  its  velocity  in  feet  per 
second;  the  pressure  p  due  to  a  column  of  fluid  is  p  =  hdg, 
where,  in  the  f.s.s  system,  p  is  expressed  in  "pounds"  per 
square  foot,  h  is  the  height  of  the  fluid  column  in  feet,  d  is 
the  density  of  the  fluid  in  slugs  per  cubic  foot  and  g  is  the  local 
acceleration  of  gravity  in  feet  per  second  per  second.  Thus  if 
the  density  of  water  is  precisely  62^  pounds  per  cubic  foot, 

it  is  exactly     —    —     slugs  per  cubic  foot,  and  the  equation 
32-r74 

p  =  hdg  gives  the  pressure  exactly  in  London  "pounds"  per 
square  foot  if  g  is  the  local  value  of  gravity.  The  numerical 
precision  which  is  required  for  most  practical  purposes  (research 
or  engineering)  does  not  justify  us  in  carefully  distinguishing 
between  the  acceleration  of  gravity  in  London,  32.1740  feet  per 
second  per  second,  and  the  acceleration  of  gravity  at  any  other 
place  on  the  earth — but  precision  of  thought  demands  that  we 
make  this  distinction.  See  page  4  of  this  volume  for  further 
discussion  of  this  subject. 

The  principle  of  the  conservation  of  energy. — Standing  before 
the  class  take  a  stone  and  lower  it  from  a  high  position  A  to  a 
low  position  B,  thus  getting  work  or  energy  out  of  the  stone; 
then  bring  the  stone  back  from  B  to  A ,  slipping  it  behind  the 
back  in  the  hope  or  pretense  of  getting  it  back  to  A  with  a  small 
expenditure  of  work!  This  procedure  will  go  far  to  give  to  the 
student  a  clear  idea  of  the  principle  of  the  conservation  of  energy 
as  discussed  in  Art.  48,  pages  68-70,  General  Physics,  and  it 
strongly  suggests  the  point  of  view  of  every  perpetual  motion 
promoter,  namely,  a  vague  and  wholly  unintelligent  expectation 
of  success  or  downright  cheating. 


HYDROSTATICS. 

27.  Experimental  demonstration  of  Pascal's  principle. — Use 

the  arrangement  described   in  Art.   51,  pages   73-74,   General 
Physics. 

Pascal's  principle  and  the  idea  of  hydrostatic  pressure.  The 
force  AF  which  is  exerted  by  a  fluid  at  rest  on  an  exposed  plane 
area  Aa  is  at  right  angles  to  Aa,  the  value  of  AF  is  the  same 
whatever  the  direction  of  Aa  (Pascal's  principle),  and  the  ratio 

AF 

—  approaches  a  definite  limiting  value  as  Aa  and  AF  approach 

zero    (AF   is  more  and  more  nearly  in  exact  proportion  to    Aa 
when   Aa  and   AF  grow  smaller  and  smaller) . 

It  is  difficult  and  in  fact  unnecessary  to  distinguish  sharply, 
in  this  outline,  between  the  elements  which  are  based  upon  or 
suggested  by  observation  and  experiment  and  the  elements  which 
are  of  a  purely  postulate  character,  but  the  outline  as  it  stands 
must  be  considered  and  fully  grasped  by  one  who  wishes  to  have 
a  clear  idea  of  the  hydrostatic  pressure  at  a  point  in  a  fluid. 

Having  established  the  idea  of  hydrostatic  pressure  at  a  point 
in  a  fluid,  it  is  easy  to  show  that  the  pressure  must  have  the  same 
value  everywhere  in  a  body  of  fluid  at  rest  when  the  fluid  is  not 
acted  upon  by  an  outside  force  like  gravity.  Consider  any 
portion  of  the  fluid  in  the  form  of  an  elongated  rectangular 
parallelepiped  or  prism.  The  forces  exerted  by  the  surrounding 
fluid  against  the  sides  of  the  prism  balance  each  other  and  these 
forces  have  no  components  parallel  to  the  axis  of  the  prism. 
Therefore  the  forces  exerted  by  the  surrounding  fluid  on  the  ends 
of  the  prism  are  equal  and  opposite,  and  consequently  the  hydro- 
static pressure  has  the  same  value  at  the  ends  of  the  prism. 

Compare  with  the  above  the  following  line  of  development, 
which  is  the  best  we  know  of  in  any  elementary  treatise  on  physics : 

(a)  It  is  pointed  out  that    AF  is  at  right  angles  to   Aa. 

52 


MECHANICS.  53 

(b)  Pressure  is  then  defined  as  force  per  unit  area. 

(c)  Then  hydrostatic  pressure  is  characterized  as  being  the 
same  in  every  direction.     (If  the  author  had  stated  that  the 
value  of    A.F    is  the  same  whatever  the  direction  of    Aa    the 
statement  would  be  intelligible,  but  it  is  non-sense*  to  say  that 
"the  pressure  is  the  same  in  every  direction.") 

(d)  Then,   after  discussing  the  non-uniform  distribution  of 
pressure  in  a  fluid  under  the  action  of  gravity,  the  author  states 
Pascal's  principle  as  follows:    "Pressure  is  transmitted  equally 
in  all  directions,  or,  if  the  pressure  at  any  point  is  increased  it  is 
increased  everywhere  throughout  the  fluid  mass  by  the  same 
amount."     The  first  clause  is  vague  and  unintelligible,  and  the 
second  clause  is  true  only  when  properly  qualified  as  to  the  time 
required  for  the  re-distribution  of  the  pressure  and  as  to  incom- 
pressibility  of  the  fluid  when  an  outside  force  like  gravity  acts 
on  the  fluid.     But  a  statement  which  has  to  be  qualified  cannot 
be  a  statement  of  a  principle.     For  example,  a  well  known 
writer  on  elementary  mechanics  illustrates  equality  of  action 
and  reaction  by  pointing  out  that  the  pull  of  a  mule  on  a  rope 
and  the  backward  pull  of  a  canal  boat  on  the  rope  are  equal 
if  the  weight  of  the  rope  is  negligible,  from  which  it  would  be 
proper  to  conclude  that  Newton's  third  law  of  motion  is  not  true 
because  ropes  always  do  have  weight ! 

The  only  comment  needed  on  the  above  statement  of  Pascal's 
principle  (?)  is  to  quote  Pascal  himself,  and  curiously  enough 
the  passage  which  best  serves  our  present  purpose  (which  is  to 
show  the  absolute  necessity  of  mathematical  thinking  in  the 
mathematical  sciences)  is  used  by  Sir  William  Hamilton  in  his 
"famous  and  terrific  diatribe  against  mathematics."  f  Speaking 

*  Let  no  one  who  is  not  familiar  with  stress  as  a  six-fold  complex  presume  to 
question  this  statement. 

t  This  phrase  is  borrowed  from  a  recent  adoration  of  mathematics  by  an 
"institutionally"  well  known  mathematician.  It  is,  of  course,  ridiculous.  See 
page  34  of  this  volume  for  further  comment,  and  consider  carefully  what  is 
meant  by  an  "institutional "  reputation  as  applying  to  an  individual.  As  applying 
to  the  present  case  we  would  change  the  word  pi  on  page  34  to  mush. 


54 


CALENDAR   OF   LEADING   EXPERIMENTS. 


Fig.  18. 


of  the  recognition  of  principles  Pascal  says  that  "Nothing  is 
wanted  beyond  a  good  sight;  but  good  it  must  be,  for  principles 
are  so  minute." 

28.  Pressure  in  a  liquid  depends  only  on  depth. — A  valve   V, 
Fig.  1 8,  is  held  up  by  a  string  which  is  attached  to  one  end  of 

the  beam  of  a  balance  scale,  and  the 
weights  on  the  scale  pan  are  adjusted 
so  that  the  •  valve  opens  when  the 
wide  vessel  W  is  filled  with  water 
to  a  certain  level  //.  The  valve  is 
then  found  to  open  when  the  narrow 
vessel  N  is  filled  to  the  same  level. 

The  use  of  engineer's  units  in 
hydrostatics.  The  use  of  the  two 
equations  (26)  and  (27)  on  page 
75,  General  Physics,  is  very  confus- 
ing. It  is  much  better  to  "go  the  limit"  and  use  f.s.s  units 
for  all  of  the  quantities  in  equation  (26),  expressing  d  in  slugs 
per  cubic  foot.  Then  equation  (27),  which  is  strictly  true  in 
London  only,  may  be  discarded.  See  discussion  of  equation 
(26)  on  page  51  of  this  volume. 

Boyle's  law. — Apparatus  for  the  verification  of  Boyle's  law  is 
familiarly  known,  but  according  to  the  authors'  experience  this 
experiment  should  be  done  by  the  student  in  the  laboratory. 

29.  Experiment  on  compressibility. — Plug  the  outlet  of  an 
ordinary  bicycle  pump.     Stike  the  pump  handle  with  a  small 
mallet  (a)  with  the  pump  barrel  filled  with  air  and   (b)  with 
the  pump  barrel  filled  with  water.     This  experiment  shows  very 
strikingly  the  enormous  difference  between  the  compressibilities 
of  air  and  water,  indeed  in  this  experiment  the  water  seems  to 
be  totally  incompressible  and  the  student  can  understand  why, 
for  most  practical  purposes,  water  is  thought  of  as  incompressible. 

Instead  of  a  bicycle  pump  with  its  slim  piston  rod,  a  one- 


MECHANICS. 


55 


inch  steel  plunger  fitting  accurately  in  a  one-inch  barrel  may  be 
used.  In  this  case  one  may  strike  the  plunger  a  heavy  blow 
without  damaging  the  apparatus. 

30.  Torricelli's  barometer. — Fill  a  clean  glass  tube  (closed  at 
one  end)  with  clean  mercury  and  invert  in  a  tumbler  of  mercury. 
Show  that  difference  of  level  of  mercury  in  tumbler  and  tube  is 
the  same  whether  the  tube  be  vertical  or  inclined— if  the  space 
above  the  mercury  contains  but  little  air. 

The  student  should  have  an  opportunity  to  take 
hold  of  the  barometer  tube  and  lift  it  off  the  bot- 
tom of  the  tumbler  to  feel  the  unbalanced  down- 
ward pressure  of  the  air  on  the  top  of  the  tube. 

31.  Archimedes'  principle. — A  solid  cylinder   C 
fits  accurately    into    the   cylindrical  pail    P,    the 
two  are  hung  from  one  pan  of  a  balance,  and  the 
weights  are    adjusted  to    give    equilibrium.     The 
cylinder    C   is  then  submerged  in  water,  and  equi- 
librium  is  again  established  by  filling  the  pail  P 

with  water.     The  buoyant  force  of  the  water  on  C         Fig.  19. 
is  equal  to  the  weight  of  the  water  in  the  pail. 

32.  Capillary  elevation  and  depression. — The  capillary  rise  of 
water  and  the  capillary  depression  of  mercury  in  small  glass 


-± 


VJ 


Fig.  20. 
Containing  water. 


Fig.  21. 
Containing  mercury. 


tubes  may  be  shown  by  placing  in  the  lantern  (horizontal  pro- 
jection arrangement)  glass  tubes  arranged  as  shown  in  Figs. 
20  and  21. 


56  CALENDAR  OF  LEADING  EXPERIMENTS. 

33.  Surface  tension  experiments.—  (a)  Mix  alcohol  and  water 
to  get  a  liquid  of  the  same  density  as  olive  oil,  place  the  alcohol- 
water  in  a  lantern  cell  and  drop  in  it  several  drops  of  olive  oil. 
Some  prefer  to  use  drops  of  carbon  bisulphide  floating  in  a  care- 
fully adjusted  salt  solution. 

(b)  Place  a  clean  horizontal  glass  plate  in  the  lantern  (vertical 
projection  arrangement).     On  the  plate  place  colored  water  to 
a  depth  of  about  a  millimeter.     Allow  a  drop  of  alcohol  to  fall 
on  the  middle  of  the  plate.     The  water  draws  itself  away  from 
the  spot  where  the  alcohol  falls  leaving  this  spot  nearly  dry 
because  the  surface  tension  of  the  pure  water  is  greater  than  the 
surface  tension  of  the  mixture  of  alcohol  and  water. 

(c)  A  fine  German  silver  wire  lies  flat  against  a  glass  plate 
which  is  placed  horizontally  in  a  lantern   (vertical  projection 
arrangement).     A  thin  layer  of  machine  oil  is  poured  on  the 
plate  and  the  oil  is  seen  to  heap  itself  up  near  the  wire  because 
of  capillary  action.     Heat  the  wire  by  an  electric  current.     The 
oil  draws  itself  away  from  the  wire  leaving  the  surface  of  the 
glass  nearly  free  from  oil  near  the  wire,  because  the  surrounding 
cool  oil  has  a  greater  surface  tension  than  the  hot  oil  near  the 
wire. 

(d)  Place   a   shallow   basin   with   glass   bottom,    thoroughly 
cleaned  and  rinsed,  in  the  lantern  (vertical  projection  arrange- 
ment).    Fill  with  clean  water.     Skim  the  surface  of  the  water 
by  blowing  a  portion  of  it  out  of  the  basin.     Scrape  a  few  very 
small  shavings  of  camphor  gum  and  allow  them  to  fall  into  the 
basin.     See  General  Physics,  page  83. 

Place  in  the  basin  above  mentioned  a  very  small  cork  boat  with 
a  split  stern  into  which  split  is  inserted  a  bit  of  camphor  gum. 
See  General  Physics,  page  83. 

Note. — The  above  mentioned  glass  basin  should  be  made  by 
cementing  two  glass  rings  to  a  glass  plate.  Cut  one  of  the  rings 
from  a  bottle  4  or  5  inches  in  diameter  and  the  other  from  a 
bottle  5  or  6  inches  in  diameter.  Grind  one  edge  of  each  ring 
flat  and  cement  to  plate  using  a  minute  quantity  of  gutta  percha 


MECHANICS.  57 

or  rubber  cement  or  marine  glue.  Then  in  the  skimming  process 
above  mentioned  the  water  that  is  blown  off  collects  in  the  annular 
space.  Both  experiments  as  above  mentioned  (and  especially  the 
experiment  with  the  cork  boat)  show  more  satisfactorily  if  the 
basin  is  heaping  full  of  water  so  that  the  water  surface  is  convex 
near  the  walls  of  the  basin. 

34.  Experiments  on  oil  flotation. — Everyone  is  familiar  with 
the  fact  that  water  does  not  spread  over  an  oily  surface.  Thus 
drops  of  water  stand  without  spreading  on  an  oily  surface 
of  glass  whereas  they  spread  out  and  cover  a  very  clean  glass 
surface.  Exactly  these  same  effects  are  shown  by  a  glass  surface 
under  water.  The  water  spreads  under  a  bubble  of  air  and 
detaches  it  from  a  clean  glass  surface  whereas  the  water  does  not 
spread  under  a  bubble  of  air  on  an  oily  glass  surface,  or,  in  other 
words,  the  bubble  of  air  clings  to  the  oily  surface.  This  may 
be  easily  shown  as  follows:  (a)  Agitate  the  water  in  a  clean 
glass  tumbler,  using  a  clean  egg  beater,  and  the  air  bubbles  all 
rise  to  the  surface  and  leave  the  walls  of  the  tumbler  clear,  (b) 
Do  the  same  in  a  tumbler  with  greasy  walls  and  numerous  air 
bubbles  are  left  clinging  to  the  tumbler  walls. 

There  seems  to  be  a  marked  difference  in  the  tendency  of  oil 
to  cling  to  small  particles  of  sand  and  to  small  particles  of  iron 
pyrite  or  zinc  blende.  Place  a  mixture  of  fine  sand  and  powdered 
iron  pyrite  into  a  tumbler  of  water,  add  a  drop  or  two  of  oil  and 
agitate  with  an  egg  beater  as  above.  The  particles  of  pyrite 
become  oily,  bubbles  of  air  cling  to  them  and  they  float,  whereas 
bubbles  of  air  do  not  cling  to  the  particles  of  sand  and  they  sink. 
This  experiment  illustrates  the  oil  flotation  process  which  is  now 
extensively  used  for  separating  particles  of  ore  from  particles  of 
sand  and  rock. 


HYDRAULICS. 

The  limitations  of  mechanics. — It  is  important  to  point  out 
to  the  student  the  fact  that  the  science  of  hydraulics  deals  almost 
entirely  with  ideal  types  of  fluid  motion,  as  explained  on  pages 
85-87,  General  Physics,  because,  in  the  study  of  any  branch  of 
science,  nothing  is  more  important  than  to  understand  to  some 
extent  the  limitations  of  the  method  used. 

The  actual  phenomena  of  fluid  motion  are  erratic  in  character, 
visibly  erratic,  and  it  is  quite  certain  that  these  phenomena  can 
never  be  correlated  by  the  classical  method  of  mechanics.* 
This  statement,  it  must  be  admitted,  represents  a  very  recent 
point  of  view,  one  that  is  accepted  only  by  a  small  group  of 
physicists,  and  we  may  be  permitted  therefore  to  set  it  forth 
more  strikingly  by  quoting  from  a  well-known  engineering 
writer  who  is  still  deeply  imbued  with  the  ideals  of  the  classical 
method  of  mechanics.  The  laws  of  fluid  motion,  he  says,  have 
not  yet  been  discovered,  although  the  movements  of  the  heavenly 
bodies  have  been  completely  formulated.  This  statement  shows 
that  the  writer  assumes  the  existence  and  expects  the  discovery 
of  rigorous  correlations  in  the  minutice  of  fluid  motion,  only, 
he  would  probably  deny  that  laws  of  fluid  motion  would  have 
anything  to  do  with  such  ephemeral  and  insignificant  phenomena. 
Every  phenomenon  that  we  contemplate  in  this  world  of  ours, 
even  such  extremely  regular  phenomena  as  the  motion  of  the 
planets,  if  examined  with  extreme  care  shows  a  substratum  of 
erratic  behavior.  In  the  case  of  fluid  motion,  however,  this 
erratic  action  rises  to  a  level  where  it  enters  into  human  values 
(if  we  may  be  permitted  to  use  that  word  to  designate  the  things 
we  consider  and  must  make  allowance  for  in  our  daily  life)  as 
exemplified  most  strikingly  in  the  phenomena  of  meteorology. 

*  Statistical  mechanics  is  indeed  not  a  branch  of  mechanics  proper.  It  is  a 
theoretical  structure  with  a  purely  postulate  basis,  and  in  its  bearing  on  upon  labora- 
tory or  research  work  it  is  a  part,  or,  indeed  the  whole  of  the  atomic  theory. 

58 


MECHANICS.  59 

Read  in  this  connection: 

(a)  Pages    119-123,   General  Physics,   where   the  method  of 
thermodynamics  is  contrasted  with  the  method  of  atomics  (the 
atomic  theory), 

(b)  Pages   322-325,   General  Physics,   where   the   method   of 
atomics  is  contrasted  with  the  method  of  mechanics,  and 

(c)  A  brief  and  very  simple  paper  on  Statistical  Physics,  in 
Science,  Vol.  XLIV,  pages  158-162,  August  4,  1916. 

35.  Turbulent    fluid    motion.     The    sensitive   flame. — Light 
an  ordinary  fish- tail  gas  jet  and  increase  the  gas  pressure  (the 
velocity  of  the  gas  in  the  jet)  more  and  more  until  the  flame 
becomes  turbulent  and  gives  off  a  roaring  sound.     This  sound 
is  produced  by  small  eddies  which  develop  in  the  border  region 
between  the  moving  gas  and  the  still  air.     In  a  high-pressure 
steam  jet  these  eddies  are  very  violent  and  numerous  and  they 
produce  the  familiar  hissing  sound  of  such  a  jet. 

When  a  gas  jet  is  on  the  point  of  changing  from  smooth  to 
turbulent  type  it  is  sometimes  very  sensitive  as  explained  on 
page  88,  General  Physics.  To  make  a  sensitive  flame  make  an 
assortment  of  nozzles  by  drawing  short  pieces  of  glass  tube  to 
points  and  cutting  off  where  the  inner  diameter  is  about  a 
millimeter.  Use  a  file  with  its  corner  ground  to  a  sharp  serrated 
edge,  and  break  off  the  points  clean  and  square.  Trying  several 
nozzles  one  usually  finds  one  that  gives  a  very  sensitive  flame 
with  ordinary  city  gas.  City  gas  pressure  is,  however,  usually 
not  great  enough  to  give  the  best  results,  and  therefore  it  is 
sometimes  advisable  to  fill  a  small  rubber  gas  bag  with  the  gas 
and  increase  the  pressure.  Natural  gas  and  gas  made  from 
gasolene  do  not  burn  satisfactorily  at  a  small  nozzle.  If  such 
gas,  only,  is  available  it  must  be  mixed  with  hydrogen  in  a  gas 
bag  or  gasometer.  Acetylene  gas  is  not  satisfactory  because 
of  the  smoky  character  of  a  large  acetylene  flame. 

36.  Instability  of  the  vortex  sheet.     The  spit-ball. — The  border 
region  between  moving  and  still  fluid  is  called  a  vortex  sheet. 


60  CALENDAR  OF  LEADING  EXPERIMENTS. 

The  instability  of  the  vortex  sheet  is  illustrated  by  the  behavior 
of  the  sensitive  flame.  The  behavior  of  the  spit  ball  as  explained 
on  page  89,  General  Physics ,  is  another  illustration.  Drop  a 
marble  in  a  tall  jar  of  water,  or  exhibit  the  zig-zag  motion  of 
small  bubbles  of  air  as  they  rise  in  water. 


DISCHARGE   RATE  OF  A  STREAM. 

In  ancient  Rome  the  rate  of  delivery  of  water  to  a  household 
was  legally  gauged  by  the  size  of  the  delivery  pipe.  It  is  evident, 
however,  that  a  small  pipe  may  deliver  water  at  any  rate  what- 
ever, depending  on  the  velocity  of  flow,  and  commercial  water 
meters  as  now  used  in  our  cities  measure  the  actual  amount  of 
water  delivered  in  a  given  time  in  cubic  feet  or  in  gallons. 

The  man  of  the  street,  or  even  the  man  of  the  farm  has  a 
very  vague  notion  indeed  of  rate  of  discharge  of  a  stream; 
his  idea  is  about  the  same  as  that  which  was  legalized  in  ancient 
Rome.  For  example,  the  rate  of  discharge  of  a  spring  is  fre- 
quently specified  as  sufficient  to  fill  a  one-inch  pipe  or  a  two-inch 
pipe,  as  the  case  may  be. 

Many  people,  however,  seem  not  to  have  any  idea  at  all  as 
to  a  rate  of  supply  of  water,  as  may  be  illustrated  by  a  recent 
discussion  of  the  water  question  in  Bethlehem,  Pennsylvania, 
where  2,000,000  gallons  per  day  is  needed.  Many  otherwise 
intelligent  citizens  proposed  a  pipe  line  from  Savior's  Lake,  the 
outflow  of  which  is  less  than  200,000  gallons  per  day;  there  is 
evidently  such  a  large  quantity  of  pure  clear  water  in  the  lake! 


61 


PERMANENT  AND   VARYING   FLOW. 

A  theoretical  discussion  of  simple  lamellar  flow  of  a  varying 
type  is  given  in  Part  VI  of  this  volume  in  connection  with  wave 
motion  in  air  pipes. 

37.  The  water  hammer. — A  striking  example  of  varying  flow 
is  afforded  by  the  well  known  water  hammer. 

When  a  water  wheel  is  supplied  through  a  long  pipe  a  great 
drop  of  pressure  occurs  at  the  wheel  when  the  water  is  turned 
on  and  while  the  water  in  the  pipe  is  being  started  (accelerated) , 
and  a  great  rise  of  pressure,  a  dangerous  rise  of  pressure,  occurs 
at  the  wheel  when  the  water-wheel  gates  are  closed.  This 
dangerous  rise  of  pressure  is  prevented,  however,  by  a  relief 
valve  which  opens  with  rise  of  pressure,  or  by  a  stand  pipe  out 
of  the  top  of  which  the  water  flows  when  the  water-wheel  gates 
are  closed. 


62 


BERNOULLI'S   PRINCIPLE. 


The  consistent  use  of  engineering  units.  Equations  (31), 
(32)»  (33)f  (34)  and  (35)  on  pages  93-96,  General  Physics,  are 
true  as  they  stand  if  foot-slug-second  units  are  used  throughout. 
That  is,  p  must  be  expressed  in  "pounds"  per  square  foot, 
d  must  be  expressed  in  slugs  per  cubic  foot  and  so  on.  See 
pages  51  and  54  of  this  volume  for  further  discussion  of  this 
point. 

38.  The  disk  paradox. — See  pages  96-97,    General   Physics. 
A  simple  form  of  this  experiment  is  to  blow  between  the  fingers 
against  a  small  piece  of  paper  lying  flat  against  the  palm  of  the 
hand. 

The  mechanics  of  the  disk  paradox  may  be  made  clear  by 
the  following  experiment.  A  jet  of  water  falls  centrally  upon  a 
thin  flat  metal  disk  in  a  basin  of  water,  and  the  disk  is  held  up 
(made  to  float)  by  the  action  of  the  jet.  Everywhere  on  top 
of  the  disk  the  water  has  high  velocity  and  low  level  (low  head 
or  pressure),  and  as  the  stream  of  water  comes  nearly  to  rest  at 
or  near  the  edge  of  the  disk  it  raises  itself  to  a  higher  level  (or 
pressure);  whereas  the  pressure  everywhere  in  the  still  water 
underneath  the  disk  corresponds  to  the  high-level  water  at  or 
near  the  edge  of  the  disk.  A 
small  needle  should  project  up- 
wards from  the  bottom  of  the 
disk  and  pass  through  a  small 
hole  in  the  disk  to  keep  the  disk 
from  moving  sidewise. 

39.  Diminution    of   pressure 
in  a  throat.— A  glass  tube  TT ', 
Fig.    22,    has   a    throat    at    a. 

The  tube  should  be  about  J  inch  bore  and  the  throat  should 

63 


64  CALENDAR   OF   LEADING   EXPERIMENTS. 

be  about  f  inch  internal  diameter.  Two  short  glass  nipples 
are  sealed  to  TT'  at  a  and  &,  a  glass  U-tube  is  connected  to 
these  nipples  by  short  pieces  of  rubber  tubing  as  shown,  and  a 
small  quantity  of  colored  water  is  placed  in  the  U-tube. 

Blowing  through  TT1  in  the  direction  of  the  arrow  draws  the 
colored  water  upwards  in  arm  a  as  indicated  in  the  figure. 
The  effect  of  friction,  alone,  would  be  to  push  the  colored  water 
downwards  in  arm  a,  because  friction  alone  would  cause  a  lower 
pressure  at  b  than  at  a. 

This  diminution  of  pressure  in  a  throat  is  utilized  in  the  jet 
pump.  Water  from  city  mains  may,  for  example,  flow  through 
TTf  and  the  pressure  at  a  may  be  low  enough  to  draw  water 
out  of  a  cellar,  the  discharge  at  T'  being  into  the  street. 

The  most  striking  form  of  jet  pump  is  the  steam  boiler  injector, 
and  the  injector  is  sometimes  operated  in  practice  by  exhaust 
steam.  The  exhaust  steam  (because  of  its  low  density)  attains  a 
very  high  velocity  when  it  reaches  the  low-pressure  region  in  the 
injector  throat.  Let  us  assume  this  velocity  to  be  1,000  feet 
per  second  for  the  sake  of  argument,  and  let  us  assume  that 
water  would  gain  a  velocity  of  100  feet  per  second  in  issuing  as 
a  jet  from  the  boiler  (boiler  pressure  about  75  "pounds"  per 
square  inch).  Then,  neglecting  friction,  water  moving  towards 
the  boiler  at  a  velocity  of  100  feet  per  second  would  carry  itself 
into  the  boiler.  But  one  unit  mass  of  steam  at  1,000  feet  per 
second  in  mixing  with  9  units  mass  of  cold  still  water  gives  100 
units  mass  of  solid  water  moving  at  a  velocity  of  100  feet  per 
second,  because  the  initial  momentum  of  the  moving  steam  must 
be  equal  to  the  total  momentum  of  the  resultant  water,  friction 
against  walls  of  throat  being  neglected. 

40.  Ball  riding  on  an  air  jet. — A  light  ball  like  a  ping-pong 
ball  is  placed  over  a  nozzle  from  which  a  blast  of  air  is  blowing, 
and  the  ball  rides  in  the  jet  without  falling  out.  One  can  with 
the  lungs  produce  a  blast  sufficiently  intense  for  this  experiment. 
Whenever  the  ball  gets  part  way  out  of  the  moving  stream  of  air 
the  higher  pressure  of  the  surrounding  still  air  pushes  it  back. 


MECHANICS. 


Fig.  23. 


41.  Ship  suction. — See  page  98,  General  Physics.  The  effect 
of  ship  suction  can  be  shown  by  supporting  two  light  balls  as 
shown  in  Fig.  23  and  blowing  between  them. 

The  discussion  of  ship  suction  in  Franklin  and 
MacNutt's  Mechanics  was  questioned  by  a  well- 
known  phsyicist,  and  to  settle  the  matter  a  letter 
was  written  to  the  Navy  Department.  The  reply, 
signed  by  the  then  Assistant  Secretary  of  the  Navy, 
stated  that  the  effect  was  unknown  to  the  depart- 
ment. It  was  added,  however,  that  the  attraction 
of  gravitation  would  pull  two  ships  together!  The 
authors  had  at  the  time  a  dim  recollection  of  the  ex- 
tensive experiments  on  ship  suction  that  had  been 
carried  out  with  great  pains  by  Naval  Constructor  (now  Admiral) 
Taylor,  and  just  before  receiving  the  above  mentioned  reply  from 
the  Navy  Department  one  of  the  authors  in  a  lecture  to  a  group 
of  school  teachers  in  Philadelphia,  had  found  three  out  of  a  total 
twenty-five  who  knew  of  ship  suction  by  experience  with  small 
boats;  and,  as  a  matter  of  fact,  the  attraction  of  gravitation 
between  two  ships  is  really  a  repulsion ! !  How  proud  we  School 
Teachers  might  feel!  if  it  were  not  largely  our  fault  that  an 
officer  of  the  Navy  (for  the  above  mentioned  reply  was  no  doubt 
formulated  by  a  navy  officer  acting  in  a  clerical  capacity)  should 
ever  come  to  substitute  the  encyclopedia  habit  for  critical 
knowledge  of  science,  and,  also  our  fault,  that  silver-spoon  youth 
should  never  develop  beyond  hopeless  dilettantism,  and  seldom 

get  even  as  far  as  that  (for  the 
then  Assistant  Secretary  of  the 
Navy  had  been  a  silver-spoon 
youth  with  all  the  tremendous 
possibilities — and  dangers — that 
go  therewith). 

42.  Flat   plate    in    a    moving 
fluid. — Figure  24  shows  the  ap- 
proximate trend  of  the  stream  lines  of  a  fluid  flowing  round  a  flat 
6 


Fig.  24. 


66  CALENDAR  OF   LEADING   EXPERIMENTS. 

plate  ab.  The  fluid  velocity  is  greater  at  a  than  at  b  so  that 
the  pressure  of  the  fluid  is  greater  at  b  than  at  a,  and  therefore 
forces  are  exerted  on  the  plate  tending  to  turn  the  plate  squarely 
across  the  stream  as  indicated  by  the  curved  arrows.  Drop  a 
small  square  of  moderately  stiff  writing  paper;  it  turns  quickly 
into  a  horizontal  position  and  rocks  back  and  forth,  and  this 
rocking  motion  causes  the  paper  to  glide  back  and  forth  sidewise 
as  it  falls.  A  flat  disk  like  a  coin  dropped  in  clear  water  is  easily 
seen  to  behave  in  the  same  way.  A  kite  with  a  plane  surface 
tends  to  rock  back  and  forth,  and  therefore  to  dart  to  and  fro 
sidewise. 

A  bit  of  paper  with  its  edges  bent  upwards  falls  steadily  like  a 
clam  shell  (with  its  convex  side  downwards)  in  clear  water,  and  a 
kite  with  a  convex  front  rides  steadily  in  the  air. 

Note. — The  lines  of  flow  in  Fig.  24  represent  the  flow  of  an 
ideal  frictionless  fluid.  In  an  actual  fluid  a  dead-water  region 
exists  behind  the  plate. 

43.  Curved  flight  of  a  spinning  ball. — Cut  corn-stalk  pith 
into  square  pieces  and  glue  them  together,  thus  forming  a  large 
block  from  which  make  a  ball  about  ij  inches  in  diameter. 
Such  a  ball  thrown  from  a  paste-board  tube  (a  mailing  tube)  by  a 
quick  sweeping  motion  rolls  along  the  side  of  the  tube  and  is 

/-^  /  0 

.S  /\  x 


Fig.  25.  Fig.  26.  Fig.  27. 

set  spinning  rapidly.  A  sharply  curved  flight  is  the  result. 
Professor  P.  G.  Tait,  using  a  rubber  balloon,  has  succeeded 
in  producing  the  three  theoretically  possible  "up"  curves  as 
shown  in  Figs.  25,  26  and  27. 

44.  The  curved  flight  of  a  high  foul-ball. — A  ball  is  set  spinning 
as  indicated  by  the  curled  arrows  c,  Fig.  28.  The  catcher  is  apt 
to  estimate  the  drop  of  the  ball  at  A  as  in  the  case  of  a  thrown 


MECHANICS. 


67 


ball  not  spinning,  but  because  of  the  rapid  spin  the  high  foul 
turns  in  towards  home  base  and  drops  at  B.  This  error  of  a 
catcher  is  sometimes  very  amusing  to  watch. 


?nd  of  bat 


Fig.  28. 


Fig.  29. 


The  curved  flight  of  a  high  foul  may  be  shown  by  shooting  a 
light  ball  marble  fashion  so  as  to  give  a  nearly  vertical  trajectory 
as  indicated  in  Fig.  29.  A  ping  pong  or  pith  ball  may  be  used, 
but  the  best  results  can  be  obtained  by  using  a  small  oak  gall. 

45.  Viscous  friction  and  eddy  friction  of  fluids.  —  It  is  perhaps 
worth  while  to  demonstrate  the  two  simplest  cases  of  -fluid 
friction  as  follows:  Measure  the  quantities  of  water  v  and  v't 
that  are  discharged  in  a  given  time  through  a  small-bore  glass 
tube,  first,  with  head  or  pressure  p  and  second  with  head  or 
pressure  pf.  Then  v/vf  =  p/p'. 

Make  a  pipe  or  duct  consisting  of  a  number  of  chambers 
separated  by  brass  disks  with  comparatively  small  holes  through 
the  disks.  Measure  the  quantities  of  water  v  and  v'  that  are 
discharged  through  the  duct  in  a  given  time,  first,  with  head  or 
pressure  p  and,  second,  with  head  or  pressure  p'  .  Then 
v/v' 


Remark.  —  In  fairly  large  pipes  eddy  friction  only  is  appre- 
ciable. At  moderate  velocities  of  flow  the  loss  of  pressure  is 
very  nearly  proportional  to  the  square  of  the  velocity.  At  very 
high  velocities  the  eddies  become  more  and  more  fine  grained 
and  the  loss  of  pressure  increases  as  vn,  where  n  is  greater 
than  2. 


68  CALENDAR  OF  LEADING  EXPERIMENTS. 

In  the  chambered  duct  as  above  described  the  eddies  are 
fixed  in  location  and  the  loss  of  pressure  is  quite  accurately 
proportional  to  z>2.  The  increasing  fineness  of  grain  of  eddy 
motion  with  increase  of  velocity  in  an  ordinary  pipe  is,  in  a  way, 
analogous  to  a  closer  and  closer  spacing  of  the  perforated  parti- 
tions in  the  above-described  chambered  duct. 


PART  II. 
HEAT. 


OPERATIVE   VERSUS    INOPERATIVE    DEFINITIONS. 

Nearly  every  physical  definition,  rightly  understood,  is  an  actual  physical 
operation.  Thus  you  define  a  cow  pasture  by  building  a  fence  around  it.  See 
page  73  of  this  volume  for  further  comment. 

We  think  of  the  flow  of  an  electric  current  through  a  wire  as  being  opposed  by 
a  kind  of  frictional  drag  or  resistance  very  much  as  the  flow  of  water  through  a 
pipe  is  opposed  by  a  frictional  drag.  Thus  we  get  the  idea  of  electrical  resistance, 
or,  rather,  the  beginnings  of  the  idea.  The  rate  of  generation  of  heat  in  a  wire  is 
proportional  to  the  square  of  the  current  flowing  through  the  wire,  and  the  pro- 
portionality factor  has  a  definite  value  for  the  given  wire.  Therefore  this  propor- 
tionality factor  is  adopted  as  the  measure  of  the  resistance  of  the  wire. 

In  every  case  a  preliminary  idea  of  a  physical  quantity  exists  before  the  quanti- 
tative definition  itself  is  established,  and  the  quantitative  definition,  rightly  under- 
stood, is  a  group  of  actual  physical  operations. 

The  preliminary  idea  of  electrical  resistance  is  expressed  in  terms  of  an  analogy 
and  therefore  it  is  easy  to  grasp  and  easy  to  talk  about,  but  the  recognition  of  an 
idea,  or  the  beginnings  of  an  idea,  in  mechanics  before  the  quantitative  definition 
is  adopted  is  difficult.  For  example,  John  Smith  knows  that  he  has  gotten  some- 
thing definite  when  he  has  secured  possession  of  a  given  batch  of  sugar,  although  he 
may  have  no  idea  as  to  how  much  sugar  it  is.  Now  the  operation  of  weighing* 
always  gives  the  same  numerical  result  when  applied  to  a  given  batch  of  sugar  and 
therefore  this  result  is  a  most  satisfactory  measure  of  quantity  of  sugar. 

The  recognition  of  heat  as  a  measurable  quantity  depends  upon  the  clear  recogni- 
tion of  the  fact  that  if  body  A  is  changed  thermally  by  the  dissipation  of  work 
and  brought  back  to  its  initial  condition  by  being  brought  into  contact  with 
another  (cooler)  body  B,  then  the  thermal  change  in  B  is  exactly  what  would  be 
produced  by  the  dissipation  of  the  original  amount  of  work  on  body  B  directly, 
so  that  the  body  A  by  virtue  of  the  thermal  change  which  is  produced  in  it  by  the 
dissipated  work  "stores"  something  which  is  equivalent  to  the  dissipated  work. 
The  complex  operation  thus  described  is  the  very  essence  of  the  quantitative 
definition  of  heat.  See  page  82  of  this  volume  for  further  comment. 

*  An  ideal  operation  of  weighing  is  usually  referred  to  in  a  brief  statement  of 
this  kind  so  as  to  avoid  the  necessity  of  referring  to  well  known  sources  of  error. 


70 


HEAT. 

Temperature  as  a  condition. — The  recognition  of  temperature 
as  a  condition  or  fact  is  explained  on  page  107,  General  Physics; 
to  find  a  temperature  go  to  a  closed  cellar  or  to  a  stuffy,  closed 
room.  A  substance  not  in  thermal  equilibrium  has,  strictly 
speaking,  no  temperature.  Touch  a  high  speed  belt  or  stick  a 
finger  into  an  electric  arc  and  you  will  be  burned,  but  the  degree 
of  hotness  cannot  be  expressed  in  either  case  as  a  temperature. 
Certain  portions  of  an  electric  arc,  the  hot  tips  of  the  carbons, 
for  example,  seem  to  be  approximately  in  thermal  equilibrium 
and  they  can  be  thought  of  as  having  a  fairly  definite  temperature, 
but  it  would  be  necessary  to  cage  a  portion  of  the  arc  gases,  keep 
them  in  a  heat-insulating  enclosure  and  wait  awhile  before  they 
could  be  said  to  have  a  definite  temperature. 

Definite  quantitative  notions  in  physics  usually  depend  upon  assumed  uni- 
formity of  distribution  in  space  and  time,  and  to  apply  such  notions  to  rapidly 
varying  conditions  the  idea  of  the  limit,  as  used  in  the  infinitesimal  calculus  must 
be  introduced.  See  pages  593-597.  General  Physics.  For  example  the  fundamental 
idea  of  density  refers  to  a  homogeneous  substance,  and  the  density  of  a  non-homo- 
geneous substance  at  a  point  is  defined  as  the  limiting  value  of  — ,  where  Am 

is  the  mass  and  Av    is  the  volume  of  a  very  small  portion  of  the  substance  at  or 
near  the  point. 

According  to  the  atomic  theory  a  portion  of  any  substance  (even  of  a  substance 
in  thermal  equilibrium)  departs  more  and  more  widely  from  thermal  equilibrium 
as  it  is  taken  smaller  and  smaller.  Therefore  the  idea  of  temperature  cannot 
be  made  to  apply  to  a  turbulent  substance  by  using  the  method  of  limits.  A 
metal  rod  which  is  red  hot  at  one  end  and  cold  at  the  other  may,  however,  be  thought 
of  as  having  a  fairly  definite  temperature  at  each  point  because  a  portion  of  the 
material  containing  many  millions  of  molecules  is  pretty  nearly  in  the  steady 
condition  which  we  call  thermal  equilibrium.  The  idea  of  temperature  is  not 
applicable  to  a  few  molecules  of  any  substance  unless  the  average  behavior  of  the 
few  molecules  during  a  long  time  is  taken  into  consideration. 

The  atomic-theory  conception  of  temperature. — The  atomic 
theory  is  a  logical  structure  which  is  built  upon  a  postulate  basis, 


72  CALENDAR  OF  LEADING  EXPERIMENTS. 

and  it  has  a  bearing  on  experimental  work  and  research  in  that 
by  its  means  very  definite  and  remarkable  ideas  can  be  developed 
and  put  to  experimental  test.  Read  in  this  connection  pages 
119-122,  General  Physics. 

The  atomic-theory  conception  of  temperature  is  entirely  apart 
from  the  things  of  experiment  (entirely  apart  from  thermo- 
dynamics), and  this  conception  has  been  fully  developed  only 
for  the  ideal  or  perfect  gas.  See  very  brief  discussion  of  the 
atomic  theory  of  gases  on  pages  325-328,  General  Physics,  in 
particular  note  the  postulate  basis  as  stated  on  page  325,  see  the 
atomic  prediction  of  Boyle's  law  as  involved  in  equation  (ii), 
page  326,  and  note  on  page  328  an  assumed  conception  of 
temperature  which  makes  our  postulated  atomic  system  con- 
form to  Gay  Lussac's  law  and  to  temperature  values  as  meas- 
ured by  the  air  thermometer. 

The  principle  of  the  equal-partition  of  energy  (which,  by  the 
way,  is  at  variance  with  many  experimental  results  and  cannot, 
therefore,  be  true)  among  the  various  degrees  of  freedom  of  a 
helter-skelter  system  seems  to  indicate  that  a  certain  tempera- 
ture means  always  a  certain  average  kinetic  energy  per  molecule ; 
but,  considering  the  probable  limitation  of  the  principle  of  equal- 
partition  of  energy  to  the  simplest  kind  of  an  ideal  gas,  it  is  by 
no  means  permissible  to  define  temperature  as  the  average 
kinetic  energy  per  molecule  of  a  substance!  In  fact  such  a 
definition  is  absurd  from  the  laboratory  man's  point  of  view, 
and  it  would  be  absurd  even  if  it  were  entirely  legitimate  from 
the  point  of  view  of  the  atomic  theory. 

Physicists  long  ago  realized  the  limited  usefulness  in  scientific 
work  of  what  may,  perhaps,  be  called  verbal  philosophy.  This 
term  unfortunately  conveys  a  suggestion  of  contempt,  but  no 
contempt  is  here  intended,  but  quite  the  opposite  of  contempt, 
for  we  refer  broadly  to  that  vital  blend  of  intellect  and  morals 
which  has  grown  up  with  the  use  of  words  in  our  age-long 
dealings  with  what  the  philosopher  calls  human  values.  Mathe- 
matical philosophy  is  the  thing,  in  the  physical  sciences.  The 


HEAT.  73 

idea  of  continuous  quantity  with  the  associated  ideas  of  func- 
tional relations  and  limits  was  the  earliest  phase  of  mathematical 
physics;  later  comes  the  widened  use  of  postulates  and  of  the 
theoretical  structures  based  thereon;  and  already  there  is  evi- 
dence of  a  new  method  (we  do  not  refer  to  statistical  physics) 
in  the  mathematics  of  discrete  things,  especially  in  the  group 
theory. 

Any  carefully  ordered  set  of  operations  of  the  analytical 
chemist  is  a  group,  the  operations  involved  in  the  making  of  a 
particular  physical  measurement  constitute  a  group,  and  a  long 
step  towards  realism  and  adequacy  of  theoretical  physics  will 
be  taken  when  the  operations  of  the  physical  sciences  are  con- 
nected in  a  mathematical  structure.  It  is  now  very  difficult 
to  think  of  these  things,  but  the  group  theory  (or  some  other 
kind  of  mathematics)  when  fully  developed  may  be  expected 
to  make  this  kind  of  thinking  easy. 

So  difficult  is  it  to  think  of  the  fundamental  operations  of  the 
physicist  that  every  teacher  of  physics  is  more  or  less  indulgenf 
and  charitable  in  his  attitude  towards  the  ever  recurring  re- 
crudescence of  verbal  philosophy  in  his  field — even  when  he 
knows  that  it  is  absurd.  Mass?  It  is  defined  as  quantity  of 
matter.  Not  by  any  means.  It  is  defined  by  the  operation  of 
weighing  by  a  balance  or  by  the  group  of  operations  involved  in 
showing  that  the  unit  of  mass  is  accelerated  m  times  as  fast  as 
the  given  body,  by  a  given  force ;  and  these  operations  are  simple 
in  comparison  with  many  of  the  measuring  operations  in  elec- 
tricity and  magnetism.  Do  you  define  a  cow  pasture  as  a 
meadow  sweet  with  clover  and  grass?  You  do  not;  you  define 
it,  for  the  cow  at  least,  by  building  a  fence  around  it! 

46.  The  Brownian  motion. — Every  student  of  physics  should 
see  the  irregular  and  incessant  to-and-fro  motion  of  very  fine 
particles  suspended  in  water,  using  a  good  microscope.  This 
motion  was  discovered  by  the  English  botanist,  Brown,  in  1827, 
and  it  is  called  the  Brownian  motion. 

Grind  a  small  amount  of  insoluble  carmine  in  a  few  drops  of 


74  CALENDAR  OF  LEADING  EXPERIMENTS. 

water,  rubbing  with  the  finger  in  a  shallow  dish.  Place  a  drop 
of  the  mixture  on  a  microscope  slide  and  use  a  magnifying 
power  of  about  400  diameters.  The  particles  in  India  ink  are 
much  finer  than  the  particles  of  carmine,  and  a  higher  magnifying 
power  is  required  to  see  them  satisfactorily. 

47.  The  expansion  of  gases. — To  show  the  very  great  expan- 
sion of  air  with  ri'se  of  temperature  place  the  long  stem  of  a  very 
dry  glass  bulb  in  a-  tall  jar  of  water  and  heat  the  bulb.  The 
expansion  drives  a  large  amount  of  air  out  of  the  bulb  as  shown 
by  the  bubbles  which  rise  in  the  tall  jar  of  water.  When  the 
bulb  cools  the  contraction  of  the  air  which  is  left  in  the  bulb  is 
shown  by  the  flow  of  water  into  the  bulb. 

The  expansion  of  air,  or,  rather,  its  increase  of  pressure  with 
rise  of  temperature  is  utilized  in  the  hot-air  engine.  A  cheap  form 
of  hot-air  engine  is  sold  by  chemical  supply  houses.  In  the  hot- 
air  engine  the  contained  air  is  shifted  from  cold  to  hot  part  of 
cylinder  and  back  again  repeatedly  by  the  motion  of  a  large 
loosely  fitting  displacement  plunger,  and  the  consequent  heating 
and  cooling  of  the  air  raises  and  lowers  its  pressure.  The  engine 
piston  moves  outwards  while  the  air  is  hot  and  its  pressure  is 
high,  and  the  energy  of  the  fly  wheel  then  pushes  the  piston 
inwards  and  compresses  the  air  while  it  is  cold  and  its  pressure 
is  low.  A  wooden  model  showing  the  large  displacement  plunger, 
the  long  cylinder  and  the  piston,  and  showing  the  red  hot  part 
of  the  cylinder  painted  red  is  a  great  help  in  the  explanation  of 
the  action  of  the  hot*air  engine. 

The  expansion  of  a  gas,  or  rather  its  rise  of  pressure  with  rise 
of  temperature  is  also  utilized  in  the  gas  engine.  In  this  case, 
however,  the  gas  which  is  used  is  a  mixture  of  inflammable 
gas  or  vapor  and  air,  and  the  rise  of  temperature  is  produced 
by  the  quick  burning  of  the  mixture.  The  toy  gas  cannon  is  the 
same  in  principle  as  the  gas  engine  and  it  makes  a  good  piece  of 
demonstration  apparatus. 

Concerning  the  gas  equation. — Several  years  ago  a  physics 
teaching  friend  sent  us  the  following  example  of  what  he  con- 


HEAT. 


75 


T 

Fig.  30. 

L   is  one  half  of  a 

tee-square. 

sidered  to  be  bad-medicine,  and  we  agree  with  him.  To  fix 
the  formula  L  =  \aP  for  the  distance  L  traveled  by  a  falling 
body  in  t  seconds  the  student  is  asked  to  remember  that  L 
is  in  fact  one  half  of  a  tee-square  as  indicated 
in  Fig.  30. 

The  unintelligent  use  of  formulas  is  cer- 
tainly a  bad  habit,  and  the  following  exam- 
ple is  taken  from  our  own  personal  experi- 
ence. It  concerns  the  familiar  gas  equation 
pv  =  RT.  A  physics  teacher  walked  to  the 
University  one  morning  in  company  with  an 
engineering  instructor,  and  the  conversation 
turned  to  the  gas  turbine,  concerning  which  the  engineering 
instructor  had  some  new  ideas  which  he  hoped  to  develop 
practically.  After  some  discussion  it  appeared  that  the  engineer- 
ing instructor  was  perplexed  concerning  a  theoretical  phase 
(as  he  would  have  expressed  it)  of  gas  action,  and  it  soon  became 
clearly  evident  that  the  perplexity  was  this:  "Since  pv  =  RT, 
how  can  a  gas  at  high  pressure  be  cold?"  The  young  man  did 
not  recognize  the  two  variables  p  and  v,  and  he  did  not  have 
the  faintest  appreciation  of  the  fact  that  this  equation  applies 
only  to  a  given  amount  of  a  gas.  He  had  allowed  the  formula 
to  take  the  place  of  the  simplest  kind  of  common  sense,  namely, 
that  anything  may  be  as  hot  or  as  cold  as  one  may  care  to  make  it. 
A  verse  from  one  of  the  Bab  Ballads  comes,  willy-nilly,  into  our 
minds  as  we  relate  this  actual  occurrence : 

"Oh,  list  to  this  incredible  tale 
Of  Thomson  Green  and  Harriet  Hale 
Its  truth  in  one  remark  you'll  sum; 

Twaddle,  twaddle,  twaddle;   twaddle  twum." 

Of  course  this  engineering  instructor  had  taken  the  regulation 
dose  of  engineering  thermodynamics  and  had  passed  it,  for  no 
one  can  become  an  engineering  instructor  who  has  not  graduated 
from  a  technical  school. 


"Es  erben  sich  Gesetz  und  Rechte  wie  eine  ew'ge  Krankheit  fort." 


76  CALENDAR  OF  LEADING  EXPERIMENTS. 

and  what  a  privilege  (Recht)  it  is  to  become  an  engineering 
instructor ! 

Nernst  has  said  that  by  far  the  most  useful  theoretical  de- 
velopments in  thermodynamics  are  those  which  are  expressed 
explicitly  in  terms  of  physical  operations,  such  as  Carnot  cycle 
operations;  and  there  are  many  other  operations  of  equal  im- 
portance. It  is  doubtful  if  any  experimental  research  in  chem- 
istry or  physics  or  engineering  has  ever  been  suggested  by  the 
elaborate  algebraic  developments  to  which  thermodynamic  theory 
unfortunately  lends  itself. 

A  perfectly  clear  understanding  of  the  physical  meaning  of 
the  second  law  of  thermodynamics  (which  is  possible  because  of 
the  postulate  character  of  the  law)  and  ability  to  interpret  an 
argument  in  thermodynamics  in  terms  of  definite  physical  opera- 
tions, these  are  the  all-important  things;  and  no  engineer, 
however  great  his  pretended  predilection  for  practicality,  can 
ever  arrive  at  these  all-important  things  when  he  starts  out  with 
a  purely  algebraic  guess  "that  heat  is  expressible  as  the  product 
of  two  factors,  one  of  which  is  entropy"  and  remains  satisfied 
with  his  algebraic  guess  as  a  definition  of  entropy ! 

All  there  is  to  engineering  thermodynamics,  and  a  great  deal 
more,  is  included  in  pages  105-177,  General  Physics,  supplemented 
by  Marks  and  Davis's  Steam  Tables.  It  may  be  that  we  should 
include,  as  an  essential  supplement,  the  presumably  unpublished 
results  of  a  recent  research  concerning  the  temperature-entropy 
diagram  for  which  an  unnamed  manufacturing  concern  is  said  to 
have  appropriated  several  hundred  dollars.*  Not  knowing  the 
results  of  this  investigation,  however,  we  are  inclined  to  point  out 
what  would  seem  to  us  to  be  the  only  proper  line  of  "research" 
in  this  matter.  Send  a  young  man  to,  let  us  say,  Berlin  to  take 
a  course  under  Nernst  and  Planck.  This  line  of  research  is  the 

*  See  a  reply  by  a  Dean  of  Engineering  to  our  statement  as  to  the  limited 
usefulness  of  the  temperature-entropy  diagram.  Bulletin  of  the  Society  for  the 
Promotion  of  Engineering  Education,  January  1917,  pages  270-274.  Our  statement 
is  to  be  found  in  the  same  Bulletin,  November  1915  pages  168-190. 


HEAT.  77 

cheapest  possible  in  this  particular  case,  it  would  lead  in  three 
years  to  perfectly  definite  results,  and  on  the  Cecil  Rhodes  scale 
it  would  cost  precisely  4,500  dollars.  It  is  not  often  that  one  is 
in  a  position  to  make  these  three  definite  specifications  regarding 
research;  and  also,  alas,  it  is  not  very  often  that  a  manufacturing 
concern  is  willing  to  appropriate  even  a  small  fraction  of  the 
necessary  cost  of  a  piece  of  research! 

48.  Northrup's  model  of  a  gas. — A  glass  jar  containing  many 
thousands  of  small  steel  balls  has  a  rapidly  moving  element  or 
agitator  at  the  bottom  which  keeps  the  balls  flying  about,  and 
the  mouth  of  the  jar  is  covered  by  a  disk  which  is  freely  sus- 
pended from  a  balance  arm.  The  flying  balls  in  the  jar  exhibit 
the  properties  of  a  gas  and  conform  to  Boyle's  law  and  to  Gay 
Lussac's  law*  as  follows : 

A  given  speed  of  the  agitator  corresponds  to  a  definite  average 
kinetic  energy  per  ball,  and  doubling  the  speed  of  the  agitator 
quadruples  the  average  kinetic  energy  per  ball.  Therefore 
square  of  speed  of  agitator  corresponds  to  or  is  proportional  to 
Kelvin  temperature  T. 

(a)  Boyle's  law. — Speed  of  agitator  being  kept  constant,  the 
force  exerted  on  the  glass  disk  is  found  by  test  to  be  inversely 
proportional  to  the  volume  of  the  space  in  which  the  balls  fly 
about  (pressure  inversely  proportional  to  volume  at  constant 
temperature) . 

(b)  Gay  Lussac's  law  as  relating  to  change  of  pressure  at  constant 
volume. — The  force  exerted  on  the  glass  disk  is  found  by  test  to 
be  proportional   to  the  square  of  the  speed  of  the  agitator, 
volume  of  space  in  which  balls  fly  about  being  constant  (pressure 
proportional  to    T  at  constant  volume). 

(c)  Gay  Lussac's  law  as  relating  to  change  of  volume  at  constant 
pressure. — The  force  exerted  on  the  glass  disk  is  found  by  test 
to  be  unchanged  in  value  when  the  square  of  the  speed  of  the 
agitator  and  the  volume  of  the  space  in  which  the  balls  fly 

*  We  refer  here  particularly  to  Gay  Lussac's  Law  as  stated  in  Art.  75,  page  113, 
General  Physics. 


78  CALENDAR  OF  LEADING  EXPERIMENTS. 

about  are  both  doubled  (volume  is  proportional  to  T  at  constant 
pressure). 

(d)  The  Brownian  motion. — A  steel  or  wooden  ball  much  larger 
than  the  small  steel  balls  or  "molecules"  is  suspended  in  the 
vessel  and  it  moves  erratically  to  and  fro  and  up  and  down  as 
the  small  balls  strike  against  it.  A  difficulty  in  this  experiment 
is  that  the  larger  ball  must  be  suspended  by  a  string,  it  tends  to 
vibrate  to  and  fro  regularly  as  a  pendulum,  and  this  regular 
pendulous  motion  is  mixed  up  with  the  erratic  motion  (the 
Brownian  motion). 

49.  Expansion  of  liquids  and  solids. — The  relative  expansion 
of  water  and  glass  may  be  shown  by  heating  a  glass  bulb  filled 
with  colored  water  which  rises  in  a  narrow  glass  tube.  It  is 
most  satisfactory  to  place  this  stem  in  the  field  of  the  projection 
lantern.  When  the  heating  flame  is  brought  against  the  bulb 
the  mean  temperature  of  the  glass  walls  is  suddenly  raised  and 
the  sudden  expansion  of  the  glass  is  shown  by  a  quick  drop  of  the 
water  level  in  the  stem.  Then  continued  heating  warms  both 
the  water  and  the  glass,  and  the  greater  expansion  of  the  water 
shows  itself  by  a  slow  rise  of  water  level  in  the  stem.  Very  few 
students  seem  to  appreciate  the  fact  that  water  contracts  as  it 
is  warmed  from  o°  C.  to  4°  C.,  and  it  is  worth  while,  therefore, 
to  repeat  this  experiment  starting  with  water  bulb  at  o°  C. 

The  irregularity  of  expansion  of  steel  in  the  neighborhood  of 
700°  C.  may  very  properly  be  shown  at  this  point  and  the 
phenomenon  of  recalescence  referred  to  later.  See  Experiment 
66. 

Several  samples  of  good  chemical  thermometers  should  be 
exhibited,  and  it  will  do  no  harm  to  state  that  a  thermometer 
bulb  and  stem  should  be  completely  immersed  in  a  fluid  or  region 
in  order  that  it  may  indicate  the  temperature  of  the  region 
accurately.  In  many  cases,  however,  the  stem  must  be  allowed 
to  project  into  the  outside  air  to  make  it  possible  to  take  a  reading. 
The  authors  know  of  a  junior  student  in  engineering  who  removed 


HEAT.  79 

a  thermometer  from  a  calorimeter  vessel  and  held  it  up  to  a  light 
so  as  to  be  able  to  read  it  accurately! 

50.  The  Trevelyan  rocker. — A  moderately  hot  soldering-copper 
is  laid  on  block  of  lead  resting  on  the  table,  and,  once  started, 
the  copper  continues  to  rock  and  produce  a  buzzing  sound.  At 
each  momentary  contact  of  the  rocking  copper  with  the  lead,  the 
lead  near  the  point  of  contact  is  heated  and  the  expansion  of  the 
lead  (a  local  swelling  too  small  to  see)  gives  a  push  to  the  receding 
copper  and  thus  keeps  up  the  rocking  motion.  The  lead  should 
present  a  cleanly  cut  surface  where  the  copper  comes  into 
contact  with  it,  and  the  copper  itself  should  be  scraped  clean. 
With  a  little  patience  this  curious 
experiment  can  always  be  made  to 
work. 


51.  Shrinking  bar  and  cast  iron  Fig  3L 
pin. — A  steel  bar  bb,    Fig.  31,  has 

a  large  pin  P  fixed  to  it  permanently,  and  a  removable  cast 
iron  pin  p.  The  middle  part  of  the  bar  bb  is  heated  nearly  to 
a  red  heat,  the  pin  p  is  put  in  place,  and  the  bar  bb  is  then 
lowered  into  position  between  two  lugs  /  and  two  lugs  /'  as 
indicated.  The  bar  bb  contracts  as  it  cools,  and  breaks  the  pin 
P. 

52.  Shrinking  of  collar  on  a  shaft. — The  important  shop  opera- 
tion of  shrinking  a  collar  on  a  shaft  makes  a  good  lecture  demon- 
stration.    A  steel  collar  turned  to  1.99  inch  inside  diameter  and 
heated  can  be  slipped  over  a  2-inch  shaft  if  one  does  it  quickly 
so  that  the  hot  collar  has  no  time  to  cool. 

53.  Thermo-stresses. — The  thin  parts  of  an  iron  casting  cool 
quickly  and  contract,  and  as  the  thicker  parts  of  the  casting 
then  cool  and  contract  excessive  stresses  are  produced  which 
often  result  in  the  fracture  of  the  casting. 

When  a  pane  of  glass  is  heated  quickly  over  a  flame  the  under 
surface  which  is  in  contact  with  the  flame  heats  and  expands, 
the  upper  portions  of  the  pane  are  subjected  to  excessive  tension, 


80  CALENDAR  OF  LEADING  EXPERIMENTS. 

and  the  pane  breaks.  Holding  the  pane  in  the  hand  one  expe- 
riences a  fairly  severe  shock  due  to  the  quick  outward  movement 
which  occurs  at  the  instant  of  rupture.  It  is  not  entirely  safe  to 
try  this  experiment. 

The  most  interesting  example  of  thermo-stresses  is  furnished 
by  the  well-known  Prince  Rupert's  drops.  Hold  the  end  of  a 
small  glass  rod  in  a  blowpipe  flame  and  allow  the  end  of  the  rod 
as  it  drops  to  fall  into  water.  Most  of  the  drops  thus  formed 
explode  before  one  can  take  them  out  of  the  water ;  but  by  trying 
patiently  and  especially  by  trying  several  varieties  of  glass  one 
can  get  drops  which  stand  indefinitely,  but  which  explode 
suddenly  when  the  surface  is  scratched  or  when  the  long  tail  is 
broken  off.  One  can  feel  a  very  decided  shock  when  a  drop 
explodes  between  the  fingers.  This  is  a  perfectly  safe  experiment 
if  one  closes  one's  eyes.  A  drop  may  be  held  in  the  field  of  the 
lantern  and  it  suddenly  disappears  on  explosion.  The  most 
satisfactory  proceeding  is  to  place  a  drop  on  a  horizontal  flat 
glass  plate  in  the  lantern  (vertical  projection  arrangement),  and 
break  off  the  tail  of  the  drop  by  means  of  a  small  pair  of  pliers. 

Fused  quartz  is  remarkable  in  that  its  contraction  when  cooled 
from  a  high  red  heat  is  not  enough  to  produce  in  it  a  stress  suffi- 
ciently great  to  produce  rupture.  A  clear  quartz  tube  may  be 
heated  in  a  blast  lamp  and  quenched  in  water  without  breaking. 

Heating  by  compression.  Cooling  by  expansion. — Any  sub- 
stance which  expands  with  rise  of  temperature  has  its  tempera- 
ture raised  by  compression  or  lowered  by  expansion.  This  effect 
is  especially  great  in  the  case  of  a  gas. 

54.  The  fire  syringe  and  Diesel-engine  ignition. — A  small 
plunger  fits  in  a  small  cylinder  and  to  the  plunger  is  attached  a 
bit  of  dry  cotton  cloth.  The  air  in  the  cylinder  is  compressed 
by  pushing  the  plunger  in  very  quickly,  and  the  cotton  is  found 
to  be  burning  if  the  plunger  is  quickly  withdrawn.  The  air  in 
the  cylinder  is  heated  by  the  compression  and  the  cotton  is 
ignited.  This  method  of  ignition  is  used  in  the  Diesel  engine. 


HEAT. 


81 


The  moving  piston  draws  in  a  cylinder  full  of  fresh  air  which  is 
then  compressed  by  the  energy  of  the  flywheel  and  heated  so 
that  an  oil  spray  injected  into  the  compressed  air  is  ignited  and 
the  burning  of  the  oil  keeps  the  temperature  (and  pressure) 
high  as  the  piston  recedes. 

55.  The  formation  of  cumulous  clouds. — On  a  quiet,  warm 
summer's  day  the  sun-heated  air  near  the  ground  frequently 
rises  in  chimney-like  columns  like  aa,  Fig.  32.  As  the  air  rises 
its  pressure  decreases  and  it  ex- 
pands. As  it  expands  its  tem- 
perature falls;  and  at  a  certain 
level  //  the  falling  temperature 
reaches  the  dew  point  and  the 
water  vapor  in  the  air  begins  to 
condense  and  form  cloud  or  fog. 
The  beautiful  dome-like  clouds  so 
commonly  seen  on  a  quiet  sum- 
mer's day  are  the  fog-filled  tops 
of  such  rising  columns  of  warm 
moist  air,  and  a  striking  characteristic  of  these  clouds  is  their 
level  bottoms. 

This  cooling  of  air  by  expansion  can  be  shown  as  follows:  A 
large  glass  flask  containing  a  little  water  is  warmed  slightly  over 
a  flame  so  that  the  contained  air  is  warm  and  moist.  Applying 
the  mouth  to  the  flask  a  portion  of  the  air  can  be  sucked  out 
(causing  a  decrease  of  pressure)  and  allowed  to  flow  back  in 
again  (causing  a  rise  of  pressure).  At  each  drop  of  pressure  the 
air  in  the  flask  is  cooled  and  the  flask  is  filled  with  cloud.  At 
each  rise  of  pressure  the  air  in  the  flask  is  warmed  slightly  and 
the  fog  or  cloud  disappears. 


Fig.  32. 


THE  CONSERVATION  OF  ENERGY  AND  THE  FIRST 
LAW  OF  THERMODYNAMICS. 

Any  definition  or  principle  in  physics  should  be  stated  so  as 
to  call  to  mind  the  operation  upon  which  the  definition  or 
principle  is  based,  or,  in  other  words,  to  suggest  trial  and  verifica- 
tion. Tell  a  woman  she  cannot  drive  a  nail  and  she  will,  if  she 
takes  you  seriously,  get  hammer  and  nail  and  try.  When  Tom 
Sawyer  tells  his  playmates  that  they  cannot  whitewash  a  fence 
they  are  all  eagerness  to  submit  the  proposition  to  an  experi- 
mental test.  But  what  young  man  ever  dreamed  of  trial  and 
verification  when  told  that  energy  can  neither  be  created  nor 
destroyed?  Indeed  the  statement  does  not  suggest  any  physical 
operation  whatever,  it  is  actually  meaningless  unless  one  knows 
beforehand  just  what  its  meaning  is  intended  to  be.  Read  in 
this  connection  Art.  48,  pages  68-70,  General  Physics. 

The  extension  of  the  principle  of  the  conservation  of  energy 
so  as  to  cover  heat  effects  (including  all  chemical  effects)  is 
called  the  first  law  of  thermodynamics,  and  stated  in  the  lan- 
guage of  experiment,  it  is  as  follows :  A  certain  change  of  state  is 
produced  in  a  substance  A  by  the  dissipation  of  work,  and  the 
substance  A  is  brought  back  to  its  initial  condition  by  contact 
with  another  cooler  in  substance  B;  then  the  change  produced 
B  is  exactly  the  same  as  if  the  original  work  had  been  expended 
(dissipated)  on  B  directly.  How  much  more  intelligible  this 
statement  than  any  abstract  statement  concerning  the  equival- 
ence of  work  and  heat ! 

Take  a  coin  and  rub  it  on  a  board,  pretending  to  heat  only  the 
coin,  then  place  the  coin  in  contact  with  a  colder  object  as  if  to 
bring  the  coin  back  to  its  initial  condition,  and  point  out  that  the 
change  which  would  thus  be  produced  in  the  colder  object  would 
be  exactly  the  same  as  if  the  original  work  had  been  expended 
on  the  colder  object  directly. 

82 


HEAT.  83 

The  molecular  conception  of  heat. — When  heat  is  imparted  to 
a  substance  the  temperature  of  the  substance  usually  rises,  but 
not  in  every  case.  Thus  when  heat  is  imparted  to  ice  at  o°  C. 
the  ice  is  converted  into  water  at  the  same  temperature.  It 
may  be  that  such  a  change  as  the  melting  of  ice  increases  the 
number  of  free  molecules  so  that  in  spite  of  a  constant  average 
kinetic  energy  per  molecule  (constant  temperature)  the  whole  of 
the  imparted  heat  goes  to  increase  the  kinetic  energy  of  molecular 
motion.  If  this  were  true  in  general  it  would  be  permissible 
to  speak  of  heat  as  the  kinetic  energy  of  molecular  motion ;  but 
it  is  not  true  in  general  and  therefore  when  heat  is  imparted  to 
a  substance  it  is  stored  in  the  substance  partly  as  an  increase  of 
molecular  kinetic  energy  and  partly,  no  doubt,  as  an  increase 
of  molecular  potential  energy  due  to  a  change  of  molecular 
configuration  of  the  substance. 

The  only  case  where  heat  imparted  to  a  substance  is  known 
to  be  stored  solely  as  increased  molecular  kinetic  energy  is  in 
the  "ideal"  or  " perfect"  gas  which  does  not  change  its  tem- 
perature during  free  expansion.  This  matter  is  discussed  very 
briefly  on  page  170,  General  Physics,  and  the  important  concep- 
tion of  the  ideal  or  perfect  gas  as  a  gas  which  conforms  to  Boyle's 
law  and  does  not  change  its  temperature  during  free  expansion  is 
explained  on  pages  171-175,  General  Physics.  Free  expansion  is 
discussed  more  at  length  on  pages  344-347  of  Franklin  and 
MacNutt's  Mechanics  and  Heat. 

Heat  of  combustion. — When  we  ask  a  young  man  in  class  to 
give  a  definition  we  always  insist — or  try  to  insist,  which,  alas 
is  not  the  same  thing — on  a  reply  which  clearly  suggests  or 
definitely  postulates  the  physical  operation  to  which  the  definition 
relates  and  in  which  it  finds  its  meaning.  Thus  we  would  wish 
a  young  man  to  say  in  defining  the  heat  of  combustion  of  coal 
that  it  is  the  amount  of  heat  you  get  out  of  a  pound  of  coal  when 
you  burn  it.  A  freshman  engineer  was  asked  to  give  this  defini- 
tion; he  had  learned  before  to  avoid  the  use  of  the  word  per  in 
the  intimate  and  more  or  less  informal  talk  of  the  classroom 


84  CALENDAR  OF   LEADING   EXPERIMENTS. 

and  he  managed  to  get  all  of  the  definition  clearly  stated  except 
to  say  that  the  coal  was  to  be  burned.  This  he  could  not  be 
led  to  say,  and  finally,  being  asked  bluntly  how  to  get  heat  out 
of  coal  he  replied  "Why,  professor,  I  don't  know!" 

This  young  man  evidently  did  not  believe  that  the  study  of  a 
mathematical  science  in  a  college  (for  even  our  technical  students 
"go  to  college"  if  you  please)  could  possibly  relate  to  so  familiar 
a  thing  as  the  burning  of  coal,  and  we  could  not  blame  him. 
For,  months  before  the  question  as  to  how  to  get  heat  out  of 
coal  was  put  to  him  he  had  been  studying  calculus  under  a 
class-room  regime  which  had  given  from  40  to  60  percent  of 
failures  year  after  year,  and  the  unintelligibility  of  the  text 
book  which  was  used  may  be  fairly  judged  by  the  following  which 
is  given  as  an  example  in  a  two-page  discussion  of  discontinuity 
(pages  22  and  23  in  fact) :  — 

"Another    form    of    discontinuity    is    seen    in    the    function 

2lfx  -{-  2 

y  =    l/x     —   when    x  =  o.     Here     y     approaches  two  limits, 

according  as  x  approaches  zero  through  positive  or  negative 
values. 

2^^x  -\-  2  2^^x    I    2 

Lim  -r£-    -  =  i.     Lim    1/x      -  =  2. 

ar=+0  2  s=-02 

We  see  that  when  x  =  o  the  curve  jumps  from  y  =  2  to 
y  =  I,  that  is  from  B  to  A  [referring  to  a  figure]." 

When  will  our  mathematics  teachers  learn  that  to  be  exacting 
and  unintelligible  is  fatal?  When  will  they  learn  to  appreciate 
the  tremendous  importance  of  appealing  to  the  intuition  or 
sense-complex  of  a  student,  especially  in  a  brief  and  necessarily 
incomplete  introductory  discussion  of  an  idea?  Any  young  man 
can  "see"  a  poor  trapped  sparrow  butt  his  head  against  a  window 
pane,  and  a  notion  of  discontinuity  based  upon  such  is  incom- 
parably more  useful  even  in  the  study  of  calculus  than  any 
purely  formal  notion,  because  the  young  man  does  not  "see" 
such  things. 


HEAT.  85 

Transfer  of  heat. — The  three  processes  by  which  heat  is  trans- 
ferred, namely,  conduction,  radiation  and  convection,  are  very 
briefly  referred  to  on  page  131,  General  Physics,  and  the  elemen- 
tary theory  of  heat  conduction  is  there  outlined. 

An  elementary  presentation  of  the  subject  of  radiation  from 
the  point  of  view  of  the  theory  of  heat  (involving  both  the 
atomic  theory  and  thermodynamics)  is  given  on  pages  301-316 
of  Franklin  and  MacNutt's  Light  and  Sound,  The  Macmillan 
Co.,  1909. 

The  transfer  of  heat  by  convection  is  exemplified  in  the  heating 
of  buildings,  and  the  purely  empirical  rules  and  equations  which 
are  used  by  heating  engineers  are  to  be  found  on  pages  653-687 
of  Kent's  Handbook. 

56.  Safety  lamp  experiment. — A  horizontal  piece  of  wire  gauze 
is  lowered  quickly  into  a  flame,  then  for  a  few  moments  the  gauze 
conducts  heat  away  from  the  flame  and  cools  the  gases  to  such 
an  extent  that  combustion  ceases  above  the  gauze.     The  flame 
stops  at  the  gauze.     Turn  on  a  Bunsen  burner,  place  a  piece  of 
wire  gauze  an  inch  or  two  above  the  burner,  light  the  gas  above 
the  gauze,  and  the  flame  burns  steadily  above  the  gauze  without 
striking  through  the  gauze.     This  action  of  wire  gauze  is  utilized 
in  the  well-known   miner's  safety  lamp,  which  was  invented  by 
Sir  Humphrey  Davy. 

57.  Cloth  singeing  experiment. — Draw  a  piece  of  fuzzy  cloth 
tightly  around  a  metal  cylinder  (a  short  piece  of  steel  shafting) 
and  run  the  flame  of  a  Bunsen  burner  quickly  over  the  cloth. 
The  body  of  the  cloth  is  kept  cool  by  contact  with  the  metal, 
but  the  fine  fuzz  is  singed  off.     In  the  finishing  of  fine  glossy 
silk  the  cloth  is  run  over  a  flame  in  close  contact  with  a  cool 
metal  cylinder. 

Wrap  a  thin  sheet  of  paper  around  a  cylinder  of  which  part  is 
solid  metal  and  part  is  wood.  Hold  the  paper-covered  cylinder 
in  the  flame  of  a  Bunsen  burner  for  a  few  moments.  The  paper 
which  is  backed  by  wood  will  be  badly  scorched  whereas  the 
paper  which  is  backed  by  metal  will  not  be  perceptibly  scorched. 


86  CALENDAR  OF   LEADING   EXPERIMENTS. 

Heat  insulation. — The  most  nearly  complete  heat  insulation  is 
that  of  the  Dewar  bulb  or  thermos  bottle,  the  action  of  which  is 
explained  on  pages  310-311  of  Franklin  and  MacNutt's  Light 
and  Sound. 

Porous  heat-insulating  materials,  such  as  saw-dust,  wool,  cork- 
board,  hair  felt  and  magnesia  pipe  covering  owe  their  heat- 
insulating  properties  to  the  low  heat  conductivity  of  the  air  which 
lies  stagnant  in  the  pores  of  the  material. 

The  phase  rule. — The  discussion  of  the  phase  rule  as  given  on 
pages  107-108,  General  Physics,  should  perhaps  be  given  in 

Chapter  VIII. 

• 

58.  Crystallization  experiments. — (a)  Moisten  a  very  clean 
glass  plate  with  a  solution  of  ammonium  chloride  and  place  it 
in  the  lantern. 

(b)  Make  a  hot  concentrated  solution  of  potassium  chlorate, 
and  allow  it  to  cool  with  occasional  brisk  stirring.  After  each 
stirring  great  numbers  of  minute  crystals  form,  and  as  these 
crystals  grow  they  settle  towards  the  bottom  as  beautiful  thin 
plates,  all  alike.  These  thin  plates  show  beautiful  colors  if  the 
vessel  is  placed  in  sunlight.  The  cooling  solution  can  be  placed 
in  a  lantern  cell  and  the  floating  crystals  projected  on  the 
screen. 

Segregation. — If  a  vessel  of  dilute  salt  solution  is  frozen,  nearly 
pure  ice  forms  at  first,  nearly  the  whole  of  the  salt  accumulates  in 
the  residual  mother  liquor  at  the  center,  and  this  residual  liquor 
then  freezes  if  the  temperature  is  sufficiently  lowered.  Something 
similar  to  this  generally  takes  place  when  any  liquid  mixture  is 
frozen.  Thus  certain  of  the  impurities  in  cast  iron  or  steel  collect 
near  the  axis  of  an  ingot.  It  is  advisable  to  exhibit  a  cut  and 
polished  section  of  a  small  ingot  of  steel  or  cast  iron  or  brass  or 
lead-antimony  alloy.  A  small  crucible  full  of  the  alloy  is  allowed 
to  cool  slowly  and  the  resulting  ingot  cut  and  polished.  The 
visible  difference  between  core  and  rim  of  ingot  is  increased  by 
etching. 


HEAT.  87 

59.  The  boiling  paradox. — A  round-bottomed  flask  with 
tightly  fitting  stopper  and  a  3O-inch  stem  contains  water,  it 
is  boiled  until  all  of  the  air  is  driven  off,  and  then  quickly  inverted 
in  a  cup  of  mercury  which  stands  in  a  deep  tray  or  tub.  See 
Fig.  33- 

Pouring  cold  water  on  the  upturned  bottom  of  the  flask, 
reduces  the  pressure  momentarily,  as  indicated  by  the  mercury 
column,  and  causes  the  water  to  boil. 


Fig.  33. 


Fig.  34. 


60.  The  cryophorus. — The  two  connected  bulbs    A    and    B, 
Fig.  34,  contain  nothing  but  water  (in     A)  and  water  vapor. 
Place  bulb   B   in  a  mixture  of  ice  and  salt,  and  the  water  in   A 
freezes.     The  water  vapor  pressure  is  kept  at  a  very  low  value 
by  the  cooling  of    B,    and  the  vaporization  of  the  water  in    A 
reduces  its  temperature  sufficiently  to  freeze  it.     Bulb  B  should 
be  stirred  round  and  round  in  the  ice  and  salt,  and  bulb     A 
should  be  occasionally  given  a  quick  blow  with  the  hand,  other- 
wise the  water  in   A    may  be  cooled  far  below  the  freezing  point 
without  freezing. 

61.  The  bursting  of  a  cast-iron  ball  by  freezing  water. — A 

cast-iron  ball  4  or  5  inches  in  diameter  with  walls  about  J  inch 
thick  is  filled  full  of  water  ,  plugged  with  a  paraffine-  or  wax- 
covered  screw,  and  placed  in  a  mixture  of  ice  and  salt. 


88  CALENDAR  OF   LEADING  EXPERIMENTS. 

A  curious  fact  is  that  a  pipe  filled  with  boiled  (air-free)  water 
is  much  more  likely  to  burst  on  freezing  than  a  pipe  filled  with 
fresh  water  (containing  air  in  solution).  As  the  fresh  water 
freezes  the  air  is  segregated,  a  line  of  air  bubbles  is  left  along  the 
axis  of  the  pipe,  and  this  line  of  air  bubbles  may  act  as  a  duct 
to  relieve  the  pressure  in  a  farther  portion  of  the  pipe.  The 
bubbles  in  ordinary  "artificial  ice"  and  most  of  the  bubbles  in 
pond  ice  are  the  air  which  is  segregated  during  the  freezing. 

62.  Regelation. — A  loop  of  fine  steel  wire  with  two  heavy 
weights  is  hung  over  a  block  of  ice,  and  in  the  course  of  an  hour 
the  wire  will  cut  through  the  ice,  but  leave  the  block  as  one  piece. 
In  front  of  the  wire  where  the  pressure  is  g*eat  the  freezing  point 
(or  melting  point)  of  the  ice  is  low,  say,   —  i°  C.     Therefore 
some  of  the  ice  (originally  at  o°  C.)  in  front  of  the  wire  melts  and 
the  temperature  of  the  region  in  front  of  the  wire  falls  to  —  1°  C. 
and  stands  at  that  temperature.     The  water  formed  by  the 
melting  flows  to  the  back  of  the  wire  where  it  is  free  from  excess 
pressure  and  where  it -stands  at  o°  C.     Therefore  heat  flows 
steadily  across  the  wire  from  the  warm  region  back  of  the  wire 
to  the  cooler  region  in  front  of  the  wire,  the  water  back  of  the 
wire  is  continuously  frozen   by  this   loss  of  heat,  and   the  ice 
in  front  of  the  wire  is  continuously  melted  by  this  supply  of 
heat. 

63.  Wood's  metal. — A  striking  example  of  low-melting-point 
alloy  is  Wood's  metal  which  consists  of  bismuth  50.1  %,  cad- 
mium 10.8  %,  lead  24.9%  and  tin  14.2  %.     Its  melting  point  is 
65.5°  C.,  whereas  the  most  fusible  of  its  constituents  is  tin  with 
a  melting  point  of  232  °  C. 

Select  a  heavy-pattern  teaspoon,  imbed  it  in  wax,  cut  away  the 
wax  even  with  one  face  of  the  spoon,  and  cover  spoon  and  wax 
with  a  block  of  plaster  of  Paris.  When  the  plaster  block  is 
hard  lift  it  off,  varnish  or  oil  its  face,  lay  the  spoon  upon  it  and 
cover  with  a  second  block  of  plaster.  When  this  second  block 
of  plaster  is  hard  lift  it  off,  loosen  and  remove  the  spoon  and  cut 


HEAT.  89 

a  gate  through  which  Wood's  metal  may  be  poured  into  the 
mould. 

64.  Bottle  washer's  experiment. — A  large  dry  bottle  with  a 
tightly  fitting  stopper  is  warmed  carefully  over  a  flame,  a  small 
sealed  glass  bulb  of  water  is  placed  in  the  bottle,  the  bottle  is 
stopped,  and  the  bulb  is  broken  by  shaking  the  bottle.     If  the 
bottle  is  at  50°  C.  the  pressure  in  the  bottle  rises  to  92  milli- 
meters above  the  atmosphere,  and  a  decided  burst  of  spray  is 
produced  when  the  stopper  is  removed. 

Transformation  points  or  temperatures. — Many  solid  sub- 
stances can  exist  in  several  states  or  modifications  as  in  the  case 
of  ice  as  briefly  described  on  page  147,  General  Physics,  the  change 
from  one  state  or  modification  to  another  state  or  modification 
takes  place  at  a  definite  transformation  temperature  (at  a  given 
pressure),  and  the  change  is  generally  accompanied  by  the 
evolution  of  heat  (as  in  the  freezing  of  water)  or  the  absorption  of 
heat  (as  in  the  melting  of  ice). 

This  matter  may  be  best  exemplified  by  a  detailed  study  in  the 
laboratory.  An  ounce  or  two  of  ammonium  nitrate  crystals  is 
melted  in  a  small  glass  beaker,  a  thermometer  reading  up  to 
200°  C.  is  placed  in  the  melt,  the  heating  flame  is  removed 
and  the  thermometer  readings  are  taken  at  intervals  of  30 
seconds  while  the  beaker  cools.  The  melt  freezes  at  168°  C., 
and  changes  its  crystalline  structure  at  125°  C.,  again  at  85°  C., 
and  again  at  35°  C. 

Retarded  transformations. — In  many  cases  a  pure  substance 
in  one  state  or  modification  can  be  heated  above  or  cooled  below 
the  transformation  temperature  at  which  it  normally  changes  to 
another  modification  before  the  actual  change  takes  place.  See 
pages  151-152,  General  Physics. 

65.  Superheating  and  undercooling  of  water. — Heat  a  small 
quantity  of  very  pure  air-free  water  in  a  clean  test  tube.     The 
water  will  rise  very  considerably  above  its  normal  boiling  point, 


90  CALENDAR   OF   LEADING   EXPERIMENTS. 

and  when  boiling  does  begin  it  takes  place  with  almost  explosive 
violence,  throwing  most  of  the  water  out  of  the  tube.  This 
experiment  succeeds  best  if  a  small  amount  of  sulphuric  acid  is 
added  to  the  water.  In  this  case  the  normal  boiling  point  is 
raised,  but  the  dilute  acid  does  not  boil  perceptibly  until  it  is 
heated  above  its  normal  boiling  point,  as  stated.  The  water  is 
freed  from  air  by  previous  boiling  in  a  larger  vessel  like  a 
flask. 

Cool  a  small  quantity  of  very  pure  air-free  water  in  a  clean 
glass  flask  by  dipping  the  flask  repeatedly  into  a  mixture  of  ice 
and  salt.  The  water  cools  5  or  6  degrees  below  o°  C.  (as  may 
be  indicated  by  a  thermometer  with  a  very  clean  bulb)  without 
freezing,  and  then  a  sudden  shock  will  cause  almost  instant 
formation  of  fine  ice  crystals  throughout  the  water. 

To  cool  water  5°  or  6°  below  o°  C.  without  freezing,  the  water 
must  be  air  free  and  extremely  clean.  It  is  difficult  to  keep  the 
water  clean  enough  in  an  open  vessel.  Therefore  this  experiment 
succeeds  best  by  using  an  ordinary  water  hammer  which  con- 
tains clean  boiled  water.  Dip  the  containing  bulb  into  slightly 
salt  water  containing  pieces  of  ice  (temperature  of  mixture 
about  —  5°  C.),  remove  carefully  and  shake.  Instead  of  a 
sealed  water  hammer  one  may  boil  distilled  water  in  a  clean 
flask  and  close  the  flask  with  a  clean  rubber  stopper  while  the 
boiling  is  taking  place,  and  a  clean-bulb  thermometer  may  pass 
through  the  stopper  to  indicate  the  temperature  of  the  under- 
cooled  water  in  the  flask. 

Superheated  and  undercooled  water  are  examples  of  retarded 
transformations.  Also  a  super-saturated  solution  of  a  salt  is  an 
example  of  a  retarded  transformation. 

Make  a  hot  saturated  solution  of  sodium  acetate  in  a  small 
clean  flask,  stop  the  flask  with  a  clean  cork  and  set  it  aside 
to  cool.  The  cool  supersaturated  solution  crystallizes  suddenly 
when  a  small  crystal  of  sodium  acetate  is  dropped  into  the  flask 
or  when  the  solution  is  touched  by  a  stick  which  has  been  wet 
with  sodium  acetate  solution  and  allowed  to  dry. 


HEAT. 


This  experiment  may  be  projected  by  the  lantern  by  using  a 
small  flask  and  placing  it  in  a  water  filled  lantern  cell. 

An  example  of  an  almost  permanently  retarded  transformation 
is  as  follows:  Make  a  strong  syrup  by  dissolving  granulated 
sugar  (or,  better,  white  rock  candy)  in  a  small  quantity  of  hot 
water,  boil  the  syrup  over  a  slow  fire  until  nearly  all  the  water  is 
driven  off,  and  set  aside  to  cool.  The  solution  does  not  freeze 
(crystallize)  at  once  but  cools  to  a  glass-like  substance.  Place 
a  small  crystal  of  granulated  sugar  on  the  surface  of  the  glass-like 
substance,  press  it  into  close  contact  therewith,  and  allow  the 
whole  to  stand  in  a  cool  place  for  several  days.  A  crystalline 
growth  will  slowly  spread  out  from  the  crystal  of  granulated 
sugar,  showing  the  slow  transformation  of  the  glass-like  substance 
(which  is  the  stable  form  at  higher  temperature)  to  a  crystalline 
form  (which  is  the  stable  form  at  lower  temperature). 

Pure  sugar  cannot  be  melted,  because  it  is  decomposed  chem- 
ically before  its  true  melting  point  is  reached,  but  nearly  all  of 
the  water  in  syrup  can  be  driven  off,  leaving  the  pure  or  nearly 
pure  sugar  in  the  melted  condition  (as  a  retarded  state  or  con- 
dition). 

66.  The  recalescence  of  steel. — A  piano  steel  wire  about  0.04 
inch  in  diameter  and  5  or  6  feet  long  is  arranged  as  indicated  in 
Fig-  35*  the  terminals  a  and  b 
being  connected  to  no- volt  sup- 
ply mains  through  a  suitable 
rheostat  and  a  controlling  switch. 
Two  small  plates  are  clamped  to 
the  wire  at  the  lower  end,  and  a 
light  wooden  pointer  catches  un- 
der a  cleat  at  c,  straddles  the 
wire  and  rests  on  the  small 
clamped  plates  as  indicated.  Fig  35 

The  movement  of  the  pointer 
while  the  wire  is  being  heated  or  cooled  shows  a  marked  irregu- 


92  CALENDAR  OF  LEADING  EXPERIMENTS. 

larity  of  expansion  at  the  transition  temperature,  which  is  about 
700°  C. 

As  the  wire  is  being  heated  its  temperature  rises  a  little  above 
700°  C.  before  the  change  of  state  or  condition  takes  place,  and 
then  when  the  change  does  take  place  a  very  noticeable  momen- 
tary drop  in  temperature  (decrease  of  brightness  of  the  hot  wire) 
occurs. 

As  the  wire  is  being  cooled  its  temperature  falls  a  little  below 
700°  C.  before  the  change  of  state  takes  place,  and  then  when 
the  change  does  take  place  a  very  noticeable  momentary  rise 
of  temperature  occurs. 

67.  Hardening  and  tempering  of  steel. — Heat  a  small  rod  of 
tool  steel  to  bright  redness,  quench  it  in  water,  and  show  that 
it  will  scratch  glass  and  that  it  is  extremely  brittle.  Explain 
the  tempering  process  as  a  more  or  less  complete  conversion  of 
the  hardened  steel  to  the  condition  which  is  stable  at  low  tem- 
peratures (soft  variety  of  steel).  This  conversion  takes  place  at 
ordinary  room  temperature  but  it  is  greatly  hastened  by  rise 
of  temperature.  A  given  degree  of  tempering  is  produced  by 
keeping  a  glass  hard  piece  of  steel  at  a  certain  temperature  for  a 
certain  time,  and  the  higher  the  temperature  the  shorter  the 
time  required  for  a  given  degree  of  tempering. 

Make  the  points  of  three  steel  tools  glass  hard,  then  make 
small  polished  areas  at  tips  of  tools,  hold  tools  one  at  a  time  with 
portions  just  back  of  tips  in  a  Bunsen  flame,  watch  the  polished 
areas  carefully,  quench  the  first  tool  when  the  polished  area 
shows  a  light  straw  color,  quench  the  second  tool  when  the 
polished  area  shows  a  brownish  color,  and  quench  the  third  tool 
when  the  polished  area  shows  a  brilliant  blue  color.  These 
colors  depend  upon  a  combination  of  time  and  temperature  and 
they  are  extensively  used  as  indicators  in  the  tempering  of  steel 
tools. 

Diffusion  and  osmosis. — When  a  solution  of  salt  is  in  thermal 
equilibrium  it  is  homogeneous,  that  is  to  say,  it  has  the  same 


HEAT.  93 

degree  of  concentration  everywhere.  When  a  solution  of  salt 
is  more  concentrated  at  the  bottom  of  a  vessel,  for  example, 
than  at  the  top,  the  inequalities  of  concentration  are  slowly 
eliminated  by  the  movement,  the  very  slow  movement,  of  the 
dissolved  salt  through  the  solution  from  the  region  of  high 
concentration  to  the  region  of  low  concentration.  This  move- 
ment of  the  dissolved  salt  is  called  diffusion. 

When  solutions  of  different  concentration  are  separated  by  a 
thin  membrane,  the  diffusion  of  the  salt  through  the  membrane 
is  hindered  more  or  less,  whereas  the  water  passes  through  more 
freely.  A  thin  bladder  which  is  filled  with  water,  tied  and  placed 
in  a  vessel  of  brine  will  become  flabby  because  of  the  loss  of 
water;  and  if  the  bladder  is  partly  filled  with  brine,  tied  and 
placed  in  a  vessel  of  water  it  will  swell  until  it  bursts  because  of 
the  entrance  of  water.  If  the  brine-filled  bladder  does  not 
burst  the  pressure  in  it  rises  to  a  fairly  definite  value,  and  this 
limiting  pressure,  reckoned  above  the  pressure  of  the  surrounding 
water,  is  called  the  osmotic  pressure  of  the  brine,  and  the  action 
of  the  membrane  in  permitting  selective  diffusion  is  called 
osmosis.  This  phenomenon  of  osmosis  which  is  a  very  widely 
occurring  phenomenon,  has  given  rise  to  the  precise  conception 
of  the  semi-permeable  membrane,  a  conception  which  is  of  great 
utility  in  the  theory  of  solutions. 

Sprinkle  salt  on  fresh  meat  or  fish,  and  the  meat  or  fish  appears 
to  become  wet.  Each  individual  cell  of  the  meat  or  fish  behaves 
like  a  bladder  filled  with  water  (which  indeed  it  is)  and  placed 
in  strong  brine.  Water  flows  out  of  the  cells  through  the  cell  walls. 

68.  Solution  and  diffusion  of  CuSO4.— Drop  a  few  crystals 
into  a  tall  jar  of  water  and  place  the  jar  where  the  class  can  see 
from  day   to  day   the  slow  diffusion  of  the  copper  sulphate 
upwards. 

69.  Osmosis. — Fill  a  bladder  partly  full  of  strong  brine,  tie 
the  bladder,  place  it  in  a  jar  of  water,  and  place  the  jar  where  the 
slow  distension  of  the  bladder  can  be  seen. 


94  CALENDAR  OF  LEADING  EXPERIMENTS. 

The  second  law  of  thermodynamics. — Next  to  the  principle 
of  the  conservation  of  energy  the  most  important  generalization 
of  physics  is  the  second  law  of  thermodynamics  or  the  principle 
of  entropy,  as  it  is  sometimes  called,  and  no  one  can  have  any 
sort  of  grasp  of  the  philosophy  of  physics  who  does  not  have 
some  degree  of  understanding  of  both  of  these  generalizations. 
The  student  should,  therefore, .be  required  to  study  very  carefully 
pages  152-169,  General  Physics. 

70.  Thermo-elastic  properties  of  rubber. — Cooling  by  expan- 
sion and  heating  by  compression  is  usually  very  small  in  liquids 
and  solids  and  difficult  to  detect.  Indeed  this  is  the  case  with 
ordinary  rubber  insofar  as  change  of  volume  due  to  change  of 
pressure  is  concerned.  But  when  a  rubber  band  is  stretched  it 
is  very  perceptibly  cooled,  and  when  the  stretch  is  relieved  the 
band  is  warmed.  This  may  be  shown  by  stretching  and  shorten- 
ing a  rubber  band  repeatedly  and  holding  the  band  in  contact 
with  the  lips. 

An  interesting  example  of  a  widely  applicable  argument  based  on  the  second 
law  of  thermodynamics  is  given  on  pages  393-396  of  Franklin  and  MacNutt's 
Mechanics  and  Heat.  Knowing  that  a  rubber  band  cools  when  stretched,  this 
argument  shows  that  a  rubber  band  stretched  by  a  hanging  weight  must  shorten 
when  heated;  and  this  conclusion  is  easily  verified  by  experiment. 


PART  III 
ELECTRICITY  AND    MAGNETISM 


95 


THE    SIDE-STEPPING   OF    MATHEMATI 


'ICS. 


The  discussion  of  Pascal's  principle  on  pages  52  and  53  of  this  volume  illustrates 
a  wide-spread  tendency  among  teachers  of  physics,  a  contempt  for  precise  thinking, 
which  is  deplorable.  What  are  we  to  do?  And  the  mathematics  teacher  also  shows 
his  contempt  for  mathematics  when  he  side-steps  mathematical  ideas  and  pins  his 
faith — and  his  students — to  formalism  and  machinery.  What  are  we  to  do  ? 
Our  own  inclination  is  that  of  P.  G.  Tait  who  in  1876  wrote  that  "In  defense  of 
accuracy  we  must  be  zealous,  as  it  were,  even  to  slaying."  But  we  thank  Professor 
Tait  for  the  humorous  turn  in  his  qualifying  clause;  for  indeed  no  human  being, 
and  least  of  all  a  timid  teacher,  would  wish  in  these  terrible  times  to  do  no  more 
than  run  amuck,  as  it  were,  among  his  friends.  See  pages  53  and  106  of  this 
volume  for  further  comment. 


MAGNETISM   AND   THE   ELECTRIC   CURRENT. 

The  traditional  plan  in  the  elementary  treatment  of  electricity 
and  magnetism  is  to  begin  with  electrostatics,  but  the  authors 
believe  that  it  is  much  better  to  begin  with  magnetism  and  the 
electric  current  because  in  this  plan  a  wide  range  of  simply 
related  experimental  facts  comes  quickly  into  view;  the  many- 
sided  magnetic  effect  of  the  electric  current  (see  pages  181,  196, 
198,  199,  209,  254  and  276,  General  Physics),  the  chemical  effect 
of  the  electric  current,  and  the  heating  effect  of  the  electric 
current. 


ONE  SET   OF   ELECTRICAL   UNITS   IS  SUFFICIENT. 

It  is  a  fact,  however  reluctant  some  of  our  physicists  may  be 
to  admit  it,  that  one  set  of  units  of  measurement  is  sufficient.  We 
make  exclusive  use,  therefore,  of  the  units  of  the  so-called 
' '  electromagnetic ' '  system. 

Some  familiarity  with  the  "electrostatic"  system  of  units  is 
demanded,  however,  by  the  wide  use  of  this  system  in  the 
literature  of  electricity  and  magnetism,  but  essential  demands 
and  conventional  demands  are  not  on  the  same  plane,  and  the 
authors  emphasize  this  fact  by  ignoring  the  "electrostatic" 
system  of  units  entirely.  It  is  sufficient  to  present  the  main 
features  of  the  electrostatic  system  of  units  orally  in  class. 


97 


THE  MYSTERIOUS  ERUPTION  OF  "V"  IN  THE  DIS- 
CUSSION  OF  THE  TWO  SYSTEMS   OF 
ELECTRICAL   UNITS. 

Any  discussion  which  places  emphasis  on  the  fact  that  the 
"electromagnetic"  unit  of  charge  divided  by  the  "electrostatic" 
unit  of  charge  is  equal  to  the  square  of  the  velocity  of  light,  but 
which  stops  short  of  a  complete  elementary  discussion  of  electro- 
magnetic wave  motion,  is,  in  our  opinion,  misleading  and  fan- 
tastic. Suppose  we  had  two  rival  systems  of  mechanical  units, 
one  of  which  was  based  on  the  arbitrary  assignment  of  unity 
as  the  density  of  the  air  and  the  other  on  the  arbitrary  assign- 
ment of  unity  as  the  compressibility  of  air,  and  suppose  that 
the  only  reference  to  sound  in  our  whole  treatment  of  physics 
were  to  mention  the  mysterious  eruption  of  v  (the  velocity  of 
sound)  in  the  ratios  of  the  units  in  the  two  systems!  would  such 
a  thing  make  for  clear  understanding?  It  certainly  would  not. 
We  have,  for  this  reason,  chosen  to  treat  the  factor  B  in  equa- 
tion (i),  page  293,  General  Physics,  as  a  simple  proportionality 
factor  to  be  determined  by  experiment,  and  it  is  exactly  that. 
Indeed  it  is  nothing  more  than  that  to  one  who  does  not  go  into 
the  theory  of  electromagnetic  wave  motion. 


98 


ELECTROMAGNETIC  WAVE  MOTION.* 

Electrical  effect  of  a  magnetic  field  which  moves  in  a  direction 
at  right  angles  to  itself;  sidewise  motion  of  magnetic  lines  of 
force.  —  The  electromotive  force  along  a  path  which  encircles 
an  amount  of  magnetic  flux  <$>  is 


where   E   is  expressed  in  ab  volts  and    -rr  is  the  rate  of  change 

of  the  magnetic  flux.  See  equation  (81),  page  256,  General 
Physics.  The  meaning  of  the  negative  sign  is  that  the  induced 
electromotive  force  is  in  the  direction  around  the  path  (or  cir- 
cuit) in  which  a  left-handed  screw  would  have  to  be  turned  to 
travel  in  the  direction  in  which  the  increasing  flux  <f>  passes 
through  the  opening  of  the  circuit  (in  the  direction  of  the  force 
with  which  the  magnetic  field  which  produces  <£  would  act  on 
a  north  pointing  magnet  pole). 
Another  form  of  equation  (i)  is 

E  =  IHv  (2) 

where  E  is  the  electromotive  force  in  abvolts  induced  in  a 
straight  rod  /  centimeters  long  which  travels  sidewise  at  a 
velocity  of  v  centimeters  per  second  across  a  magnetic  field  of 
which  the  intensity  is  H  gausses  as  indicated  in  Fig.  184,  page 
255,  General  Physics.  See  equation  (80),  page  255,  General 
Physics. 

Let  us  think  of  the  rod  be,  Fig.  184,  page  255,  General  Physics, 
as  stationary  and  imagine  the  magnetic  field  as  sweeping  past 

*  This  discussion  is  not  intended  to  be  entirely  rigorous,  or  complete.  Indeed 
the  idea  of  sidewise  motion  of  identifiable  lines  of  force  is  highly  artificial.  A 
better  point  of  view  is  that  in  which  the  differential  equations  of  wave  motion  are 
derived  from  the  fundamental  circuit-equations  (i)  and  (8)  below.  This  point  of 
view  is  developed  in  Part  VI  of  this  volume. 

99 


IOO 


CALENDAR  OF   LEADING  EXPERIMENTS. 


the  rod  at  velocity  equal  and  opposite  to  v.  Then  equation  (2) 
still  applies,  E  is  the  electromotive  force  in  abvolts  along  the 
stationary  rod,  and  this  electromotive  force  exists  whether  the 
rod  is  there  or  not  so  that  E/l  is  the  electric  field  intensity  in 
abvolts  per  centimeter  produced  by  the  sidewise  motion  of  the 
magnetic  field.  Therefore,  representing  this  electric  field  inten- 
sity by  /  and  remembering  that  v  has  been  reversed  in  sign, 
equation  (2)  becomes 

f=-vH  (3) 

Magnetic  effect  of  an  electric  field  which  moves  in  a  direction 
at  right  angles  to  itself;  sidewise  motion  of  electric  lines  of 
force. — A  charged  condenser  A  A  BB,  Fig.  36,  is  allowed  to 


B 


edgewise  view 


side  view 


Fig.  36. 


discharge  through  a  wire.  The  current  /  is  equal  to  the  rate 
of  decrease  dQ/dt  of  the  charge  on  either  plate.  But  Q  =  CE, 
where  E  is  the  electromotive  force  between  the  condenser  plates 
in  abvolts  and  C  is  the  capacity  of  the  condenser  in  abfarads, 
and 

—  ;;•;•;;•;      .         .--.        C  =  B±  (4) 

where  a  is  the  area  of  one  face  of  a  condenser  plate,  x  is  the 
distance  between  the  condenser  plates  and  B  is  a  proportionality 
factor  whose  value  is  to  be  determined.  See  Art.  206,  page  293, 


ELECTRICITY  AND   MAGNEtlSM.    ' 


General  Physics.     Therefore,  using  the  value  of  C  from  equation 
(4)  in  the  expression    Q  =  CE,   we  get 


-  (5) 

36 

But  E/x  is  the  intensity  of  the  electric  field  in  the  air  space 
between  the  condenser  plates,  and  a  X  E/x  is  the  electric 
flux  ^  from  plate  to  plate.  Therefore 


Q  =  B*  (6) 

Differentiating  we  get 


Now  the  magnetomotive  force  M  along  any  path  which 
encircles  /  abamperes  of  current  is  equal  to  4^7*.  Therefore, 
using  M/4TT  for  I,  equation  (7)  becomes 

M  =  ^B-d-jt  (8) 

The  magnetomotive  force  may  be  thought  of  as  being  taken 
along  the  dotted  line  in  the  side  view  in  Fig.  36.  A  changing 
electric  flux  through  the  opening  of  any  loop  always  means  the 
existence  of  a  magnetomotive  force  around  the  loop  in  accordance 
with  equation  (8)  which  is  identical  in  form  to  equation  (i) 
except  for  the  presence  of  the  proportionality  factor  4irB  and 
the  absence  of  the  negative  sign  [the  positive  sign  is  proper  in 
equation  (8)  because  the  magnetic  field  surrounding  an  electric 
current  is  in  the  direction  in  which  a  right-handed  screw  would 
have  to  be  turned  to  travel  in  the  direction  of  the  current]. 
Considerations  almost  identical  in  form  to  those  involved  in  the 
discussion  of  Fig.  184,  page  255,  General  Physics,  would  lead  to 
an  equation  formally  identical  in  form  to  equation  (3),  namely 

H  =  4*Bvf  (9) 

where    H    is  the  intensity  in  gausses  of  the  magnetic  field  pro- 

*  See  pages  79-81,  Advanced  Electricity  and  Magnetism.  Franklin  and  MacNutt, 
The  Macmillan  Co.,  1915. 


ioi2v          CALENDAR  OF  LEADING  EXPERIMENTS. 


duced  by  the  sidewise  motion  at  velocity    v    of  an  electric  field 
of  intensity  /  abvolts  per  centimeter. 

Meaning  of  opposite  signs  in  equations  (3)  and  (9). — Con- 
sidering that  we  have  chosen  to  think  of  the  magnetic  field 
moving  to  left  in  Fig.  184,  page  255,  General  Physics,  as  being 
exactly  equivalent  to  motion  of  rod  be  to  the  right,  and  knowing 
that  induced  electromotive  force  is  from  c  towards  b  when  the 
magnetic  field  is  directed  away  from  the  reader,  it  is  evident  that 
the  relation  between  /,  v  and  H  as  expressed  by  equation  (3) 
must  be  as  shown  in  Fig.  37. 

Multiplying  H  by  —  v  in  equation  (3)  turns  H  in  a  clock- 
wise direction  as  seen  from  the  left  in  Fig.  37. 

Multiplying  /by  -f  v  in  equation  (9)  turns  /  in  a  counter- 
clock-wise  direction  as  seen  from  the  left  in  Fig.  38. 

The  electromagnetic  plane  wave. — It  is  convenient  to  turn 
Fig.  38  so  that  directions  of    v,  H    and   /    are  the  same  as  in 
-  37 1  thus  giving  two  identical  figures,  39  and  40. 


Fig.  37. 

H  is  perpendicular  to  the  plane  of 
the  paper  and  directed  away  from 
reader.  This  figure  shows  the  relation 
between  /,  v  and  H  as  expressed  by 
equation  (3). 


O 


Fig.  38. 

/  is  perpendicular  to  the  plane  of  the 
paper  and  directed  towards  reader. 
This  figure  shows  the  relation  between 
H,  v  and  /  as  expressed  by  equation 
(9). 


Now  it  is  evident  that  the  /  which  is  produced  by  sidewise 
motion  of  H  moves  along  with  H  because  the  induced  /  is 
always  where  H  is;  and,  similarly,  the  H  which  is  produced 
by  sidewise  motion  of  /  moves  along  with  /.  Therefore: 

(a)  The  H  in  Fig.  40  which  is  produced  by  the  sidewise  mo- 
tion of  /  may  be  the  H  in  Fig.  39,  and 


ELECTRICITY  AND   MAGNETISM. 


103 


(b)  The  /  in  Fig.  39  which  is  produced  by  the  sidewise  motion  of 
H  may  be  the  /  in  Fig.  40;  v  being  the  same  in  both  figures. 


Fig.  39. 

/   is  the  electrical  field  produced  by 
sidewise  motion  of   H   at  velocity   v. 


Fig.  40. 

H   is  the  magnetic  field  produced  by 
the  sidewise  motion  of  /  at  velocity   v. 


Two  such   mutually  sustaining  magnetic  and  electric  fields 
traveling  along  together  constitute  a  plane  electromagnetic  wave. 

A  front  view  (as  seen  from  v)  of  Figs. 
39  and  40  is  shown  in  Fig.  41,  in  which 
the  full  lines  represent  magnetic  lines 
of  force  and  the  dotted  lines  represent 
electric  lines  of  force. 

It  is  evident  from  what  is  stated 
above  that  equations  (3)  and  (9)  are 
to  be  thought  of  as  simultaneous  equa- 
tions inasmuch  as  /,  H  and  v  have 
identically  the  same  meanings  in  both 
equations.  Therefore,  multiplying  equa- 
tions (3)  and  (9)  member  by  member 
we  get 


i 

;  t!    • 

{ 

A 

; 

"!   "I    i 

i  —  (_  . 

i 

/ 

^  ! 

K  i 

j    j    |    ' 

I 

k 

r,  J    i 

i 

)!   '  .1 

i 

[ 

i 

•  i 

r 

u 

!  I 

Fig.  41. 

Front  view  of  plane  elec 
tromagnetic  wave  (coming 
towards  reader).  Full  lines 
represent  magnetic  lines  of 
force,  dotted  lines  represent 
electric  lines  of  force. 


or,  ignoring  the  negative  sign  which  here  relates  to  the  quaternion 
interpretation  of  the  square  of  a  vector  quantity,  we  get 


V  = 


(II) 


Also,  dividing  equation  (3)  by  equation  (9)  member  by  member 
and  (ignoring  negative  sign  as  above)  we  get 


/ 

H 


(12) 


104  CALENDAR  OF  LEADING   EXPERIMENTS. 

Equation  (n)  gives  the  velocity  at  which  two  mutually  sustain- 
ing electric  and  magnetic  fields  must  travel,  and  equation  (12) 
gives  the  necessary  ratio  of  the  two  mutually  sustaining  fields. 
The  value  of  v  is  2.99778  X  io10  centimeters  per  second  in  air 
(the  velocity  of  light),  and  therefore  the  value  of  '  B  is 
8.84  X  io~23  so  that  the  capacity  of  an  air  condenser  in  abfarads 
as  expressed  by  equation  (4)  is 


abfarads 


(l3) 


THE  ELECTROSTATIC  UNIT  OF  CHARGE. 

According  to  equation  (6)  in  the  above  discussion  the  electric 
flux  which  emanates  from  Q  abcoulombs  of  charge  (comes  in 
to  the  charge  if  it  is  negative)  is  equal  to  Q/B.  This  relation  is 
general,  and  we  may  apply  it  to  a  concentrated  positive  charge 
Qr.  The  electric  flux  passes  out  from  Qf  symmetrically  in  all 
directions.  Let  /  be  the  electric  field  intensity  at  a  point  distant 
r  from  Q'.  Then  4?rr2/*  is  the  electric  flux  from  Qf  so  that 


or 


Let  another  charge   Q"   be  placed  at  a  point  distant   r   from 
Q'.     Then,  according  to  equation  (95),  page  308,  General  Physics  t 

i     O'O" 
fQ"  or  —  n'^T~  >   is  the  force   F  in  dynes  exerted  on   Q"  by 

47T-O         T 

Q'.    That  is 


r* 

*  Argument  similar  to  Art.  135  page  135  General  Physics. 


ELECTRICITY  AND   MAGNETISM  105 

Consider  two  equal  concentrated  charges  g  (expressed  in  ab- 
coulombs)  which  exert  a  force  of  one  dyne  on  each  other  when  they 
are  at  a  distance  of  one  centimeter  apart.  Putting  F  =  I  and 
r  =  I  inequation  (15),  and  writing  qq  or  qz  for  Q'Q"j  we  have 


or 

g  =  A/4^B  (16) 

But  A/47TJ3  =  i/v  according  to  equation  (n)  so  that  equation 
(16)  becomes 

g  =  -  abcoulombs  (17) 

The  charge  g  as  here  specified  is  the  electrostatic  unit  of  charge, 
and  therefore  there  are  i/v  abcoulombs  in  one  electrostatic 
unit  of  charge. 


POTENTIAL. 

The  velocity  potential  of  a  fluid  at  a  point  (when  it  exists) 
is  the  height  at  that  point  of  an  imagined  hill  whose  slope  is 
everywhere  equal  to  the  fluid  velocity.  For  example,  consider 
a  layer  of  fluid  lying  on  a  plane  and  moving  at  constant  uniform 
velocity  v  in  the  direction  of  the  #-axis  of  reference.  Then  if 
\f/  is  the  height  of  the  potential  hill  at  a  point  whose  abscissa 

d\l/ 
is   x  we  have   —  =  v,   whence   ^  =  vx  +  a  constant. 

There  are  distributions  of  fluid  velocity  which  have  no  velocity 
potential ;  for  example,  a  layer  of  water  on  a  rotating  disk. 

The  potential  of  an  electric  field  at  a  point  (when  it  exists) 
is  the  height  at  that  point  of  an  imagined  hill  whose  slope  is 
everywhere  equal  to  the  electric  field  intensity.  For  example, 
the  electric  potential  is  ^  =  fx  +  a  constant  in  a  uniform  electric 
field  of  intensity  /,  the  x-axis  being  chosen  in  the  direction  of 
the  field.  If  /  is  expressed  in  volts  per  centimeter  and  x  in 
centimeters,  the  height  of  the  potential  hill  is  expressed  in  volts. 

The  idea  of  potential  is  useful  because  the  precise  mode  of 
distribution  of  an  electric  field  (or  of  a  magnetic  field  or  fluid 
velocity)  in  space  may  be  easily  thought  of  and  easily  formulated 
in  terms  of  the  potential  distribution,  and  especially  because 
certain  mathematical  transformations,  such,  for  example,  as  are 
involved  in  a  change  of  axes  of  reference,*  are  easily  made 
when  the  field  distribution  is  expressed  in  terms  of  potential. 

We  carefully  avoid  the  use  of  the  word  potential  in  our  General 
Physics  for  three  reasons,  namely,  (a)  Because  electric  field 
(or  magnetic  field  or  fluid  velocity)  is  the  physically  real  thing, 
and  potential  is  merely  a  mathematical  idea,  (b)  Because  the 

*  A  good  example  of  such  a  transformation  is  given  on  page  246  of  Franklin, 
MacNutt  and  Charles's  Calculus. 

106 


ELECTRICITY  AND   MAGNETISM.  107 

idea  of  potential  is  useful  only  when  one  is  considering  the  space 
distribution  of  a  field,  and  such  matters  are  largely  beyond  the 
scope  of  our  General  Physics,  and  (c)  Because  the  common  use 
of  the  term  potential  is  hopelessly  vague  and  shot  through  and 
through  with  absurdity  from  the  rigorous  mathematical  point 
of  view.  We  believe  in  physical  reality,  and  we  believe  in 
mathematical  thinking;  therefore  we  cannot  tolerate  the  common 
use  of  the  word  potential. 

Let  no  one  imagine  that  by  common  usage  we  refer  to  the 
habit  of  the  man  in  the  street.  Recently,  in  discussing  the 
question  of  potential  with  a  teacher  of  physics  in  a  technical 
school,  he  said  "Potential,  potential;  you  mean  the  potential 
function."  Exactly  so;  but  his  implication  was  that  he  always 
meant  something  very  much  simpler  than  the  '  potential  function ' 
when  he  used  the  term!  At  a  given  time  temperature  has  a 
definite  value  at  every  point  on  the  map,  and  even  a  complex 
minded  mathematician  would  be  shocked  if  his  inquiry  as  to  the 
temperature  at  Cape  Cod  were  to  be  met  by  the  response 
" Temperature,  temperature;  you  mean  the  temperature  func- 
tion," and  he  might  reply,  somewhat  impatiently,  "Yes,  in  the 
name  of  all  that  is  precise  and  definite,  that  is  what  I  do  mean." 

It  is  very  greatly  to  be  desired  that  our  teachers  of  physics 
and  electrical  engineering  should  understand  the  essentials  of 
what  is  nowadays  called  vector  analysis,  the  mathematics  of  the 
distribution  of  scalar  and  vector  values  in  space.  A  very  simple 
discussion  of  this  general  subject  is  given  on  pages  210—253  of 
Franklin,  MacNutt  and  Charles'  Calculus.*  The  idea  of  electric 
potential  and  its  use  in  the  study  of  the  electric  field  is  discussed 

*  This  book  is  recommended  by  a  large  western  university  for  use  by  its  students 
in  correspondence  courses  for  private  study,  it  has  been  used,  and  to  some  extent 
successfully,  in  the  night  school  and  in  the  high  school  and  in  the  college,  and  it 
has  been  highly  praised  as  a  good  book  for  the  graduate  student.  To  such  a  state 
of  topsy-turvydom  has  our  vague  and  unintelligible  collegiate  instruction  in  the 
mathematical  sciences  brought  us!  Any  mathematical  treatise  that  is  intelligibly 
straightforward  and  definitely  clear  and  precise  is,  it  would  seem,  good  medicine 
in  our  time  for  anybody. 


108  CALENDAR  OF   LEADING  EXPERIMENTS. 

at  length  on  pages  165-192  of  Franklin  and  MacNutt's  Advanced 
Electricity  and  Magnetism,  The  Macmillan  Co.,  1915. 

71.  The  electromagnet. — An  experiment  which  is  very  striking 
is  to  exhibit  a  large  and  powerful  electromagnet.     To  eliminate 
the  spectacular  features,  however,  let  the  magnet  be  of  moderate 
size,  consisting  of  a  straight  bar  of  iron  or  mild  steel  with  a  coil 
of  wire  upon  it.     Show  that  the  bar  loses  nearly  all  of  its  magne- 
tism when  the  exciting  current  is  reduced  to  zero  by  opening  the 
supply  switch.     Magnetize  a  bar  of  hardened  steel  by  placing 
it  in  the  coil  and  show  that  it  retains  a  large  part  of  its  magnetism 
when  removed  from  the  coil. 

72.  The  heating  effect  of  the  electric  current,  although  al- 
ready familiar  to  every  student,  may  be  strikingly  shown  by 
stretching  an  iron  or  German  silver  wire  across  the  lecture  table 
and  connecting  it  through  a  supply  switch  to  no- volt  supply 
mains. 

73.  The  chemical  effect  of  the  electric  current  may  be  shown 
by  placing  two  lead  electrodes  in  a  jar  of  dilute  sulphuric  acid 
and  connecting  them  through  a  heavy-current  rheostat  to  direct- 
current  supply  mains,  or  a  thin  coating  of  copper  may  be  de- 
posited on  a  bright  piece  of  platinum  or  silver  foil  using  a  solution 
of  copper  sulphate  as  the  electrolyte  and  using  a  current  of  not 
more    than,  say,  half   an  ampere;  an    ordinary  glow  lamp  in 
circuit  will  provide  for  a  suitable  current  from  no- volt  direct- 
current  mains.     See  Experiment  84. 

74.  The  magnetic  compass. — The  most  convenient  form  of 
demonstration  compass  is  a  magnetized  steel  needle  six  or  eight 
inches  long  mounted  on  a  pivot  stand. 

75.  Poles  of  a  magnet. — Dip  a  bar  magnet  into  a  box  of  iron 
filings  and  point  out  that  filings  cling  only  to  end  parts  of  bar 
as  shown  in  Fig.  114,  page  187,  General  Physics. 

76.  Attraction  of  unlike   magnet  poles.     Repulsion  of  like 
magnet  poles. — Most  young  men  are  familiar  with  magnetic 


N 


V 


handle^ 


S 


ELECTRICITY  AND   MAGNETISM.  109 

attraction,  whereas  but  few  are  familiar  with  magnetic  repulsion. 
Mark  north-pointing  ends  of  two  compass  needles  by  painting 
them  red.  Remove  one  of  the  needles  from  its  pivot  and  show 
that  its  red  end  repels  the  red  end  of  the  other 
needle  and  attracts  the  unpainted  end  of  the 
other  needle. 

Place  a  short  straight  cylindrical  steel  mag 
net    NS,    Fig.  42,  on  a  horizontal  glass  plate 
in    the   lantern    (vertical    projection  arrange-          Fig  42 
ment)  and  bring  another  magnet     N'S'    near 

to  it,  as  shown.     In  one  position    NS    is  repelled  by    N'S', 
and  in  the  reverse  position    NS  is  attracted. 

77.  Distributed  poles  and  concentrated  poles. — Magnetize  a 
short  thick  bar  and  a  long  slim  bar  of  hardened  steel,  as  explained 
in  experiment  71.  Dip  both  magnets  into  a  box  of  iron  filings 
and  point  out  that  filings  cling  to  extended  portions  near  the 
ends  of  the  short -thick  bar,  but  only  to  short  portions  near  the 
ends  of  the  long  slim  bar. 

Magnetize  a  long  slim  steel  bar,  remove  it  from  the  magnetizing 
coil,  turn  it  end  for  end,  and  bring  one  end  of  the  bar  thus 
reversed  into  the  end  of  the  coil.  Dip  the  bar  into  filings  and  a 
widely  distributed  pole  will  be  shown  to  be  spread  over  the  middle 
portions  of  the  bar,  with  two  moderately  concentrated  poles  at  its 
ends.  Show  by  using  a  compass  needle  that  the  end  poles  are 
alike  in  kind  and  that  the  middle  pole  is  unlike  the  end  poles. 

Remark. — It  may  be  explained  to  the  student  that  the  idea 
of  the  concentrated  magnet  pole  is  a  differential.  Thus  the 
element  of  force  action  between  any  two  magnet  poles  is 

dm' -dm"                          C  Cdrn'-dm" 
AF  = so  that  ^=11 ,  and  this  equation 

is  a  sextuple  integral  although  set  in  the  form  of  a  double  integral. 
Of  course  it  does  no  good  to  explain  the  concentrated  pole  to  the 
student  as  a  differential,  and  our  easy  reference  to  double  and 
sextuple  integrals  is  made  to  deter  the  too  eager  teacher  from 
making  the  attempt. 


no 


CALENDAR  OF   LEADING   EXPERIMENTS. 


78.  The  unit  pole. — It  is  a  great  help  to  the  understanding  of 
Art.  127,  page  188,  General  Physics,  to  have  a  number  of  pairs 
of  long  slim  magnets;  taking  one  pair  in  the  hands  at  a  time,  hold 
them  so  that  two  of  their  poles  are  about  one  centimeter  apart 
and  the  other  two  poles  very  far  apart  (assumed  to  be  indefinitely 
far  apart),  pretend  to  estimate  the  force  of  attraction  or  repulsion. 
The  student  will  thus  see  that  it  is  possible  to  find  a  pair  of 
magnet  poles  which  attract  or  repel  each  other  with  a  force  of 
one  dyne  when  they  are  one  centimeter  apart. 

79.  Magnetic  figures. — Place  a  short  bar  magnet  on  a  hori- 
zontal glass  plate  in  the  lantern  (vertical  projection  arrangement), 
place  a  thin  glass  plate  over  the  magnet  with  steadying  wedges 
or  blocks  under  its  edges,  dust  fine  iron  filings  very  sparingly  on 
the  thin  glass  plate  and  tap  the  thin  glass  plate  lightly  with  a 
hard  object  like  a  short  glass  tube.     It  is  desirable  to  show  also 
the  filaments  of  filings  between  two  opposite  poles  of  two  broad 
ended   magnets   as   indicated   in   Fig.    119,   page    191,   General 
Physics. 

80.  Behavior  of  a  magnet  in  a  non-uniform  magnetic  field. — 

The  two  poles  of  a  magnet  which  is  placed  in  a  uniform  magnetic 

field  (a  field  which  has  everywhere 
the  same  intensity  and  of  which 
the  lines  of  force  are  parallel)  are 
acted  upon  by  equal  and  opposite 
forces,  as  indicated  by  the  arrows 
in  Fig.  120,  page  191,  General 
Physics.  Therefore  a  uniform 
magnetic  field  tends  only  to  turn 
a  magnet  into  a  certain  direction, 
the  direction  of  the  field.  The 
two  poles  of  a  magnet  which  is 
placed  in  a  non-uniform  magnetic 
field  (a  field  which  does  not  have  everywhere  the  same  intensity 
and  of  which  the  lines  of  force  are  not  parallel)  are  not  equal  and 


Fig.  43. 


ELECTRICITY  AND   MAGNETISM. 


Ill 


opposite,  and  they  tend  not  only  to  turn  the  magnet  into  the 
direction  of  the  field  but  also  to  impart  to  the  magnet  a  motion 
of  translation.  Thus  the  two  arrows  F  and  F'  show  the  two 
forces  exerted  on  the  small  magnet  ns  in  Fig.  43. 

The  attraction  of  a  particle  of  iron  by  a  magnet  depends  in  the 
first  place  on  the  conversion  of  the  particle  into  a  magnet,  and  in 
the  second  place  on  the  non-uniformity  of  the  magnetic  field  in 
which  the  particle  finds  itself. 

The  magnetic  field  near  the  flat  end  of  a  large  magnet  pole  is 
nearly  uniform  as  indicated  by  the  lines  of  force  in  Fig.  44; 


S^^fSSS*S85  yj  i!  i^'W'^w^i^f^' 


Fig.  44. 


Fig.  45. 


near  the  sharp  corners,  however,  the  field  is  distinctly  non- 
uniform.  Therefore  small  particles  of  iron  are  not  perceptibly 
attracted  by  the  flat  face  of  the  pole,  but  only  by  the  sharp 
corners.  Pass  the  square  end  of  a  magnet  over  a  table  on 
which  fine  iron  filings  are  dusted  very  sparingly,  and  the  filings 
are  gathered  only  on  the  sharp  corners  of  the  pole.  The  lines 
of  force  in  the  neighborhood  of  a  sharply  pointed  pole  diverge 
strongly  as  indicated  in  Fig.  45,  that  is  to  say,  the  magnetic 
field  in  the  neighborhood  of  the  point  is  non-uniform  to  a  high 
degree,  and  a  pointed  pole  exerts  a  strong  attraction  for  a  small 
particle  of  magnetic  material.  The  eye-magnet  which  is  used 
for  removing  particles  of  steel  from  the  eye  has  a  sharp  point, 
and  the  essential  feature  of  the  magnetic  ore  separator  as  used 
to  separate  the  particles  of  magnetic  material  from  finely  crushed 
ore  is  a  sharply  pointed  magnet. 


112 


CALENDAR  OF  LEADING  EXPERIMENTS. 


81.  Oersted's  experiment. — Connect  a  flexible  wire  (one  strand 
of  a  lamp  cord)   to   no- volt  direct-current  mains  through  a 
rheostat  so  as  to  have  a  current  of  about  2  or  3  amperes  in  the 
wire.     Stretching  a  portion  of  the  wire  in  a  north-south  direction 
between  the  hands,  bring  it  down  over  a  compass  needle  and 
note  the  deflection  of  the  needle.     Interpret  this  experiment  as 
explained  in  Art.  137,  page  197,  General  Physics.     Connect  a 
copper  plating  arrangement  in  the  circuit  so  as  to  deposit  copper 
on  a  bright  platinum  or  silver  surface,  and  point  out  the  fact  that, 
according  to  the  indication  of  the  magnetic  needle,  the  current 
flows  through  the  copper  sulphate  solution  towards  the  platinum 
or  silver  electrode  on  which  copper  is  deposited. 

Wrap  the  flexible  wire  several  times  around  a  small  soft  iron 
bar,  test  the  magnetic  polarity  of  one  end  of  the  bar,  and,  having 
previously  established  the  direction  of  flow  of  current  through 
wire  as  above  explained,  point  out  the  meaning  of  Fig.  130,  page 
198,  General  Physics. 

82.  Side  push  of  the  magnetic  field  on  an  electric  wire. — (a) 
Stretch  a  6  or  7-foot  span  fine  copper  wire  loosely  in  front  of  a 
strong  electromagnet  as  indicated  in  Fig.  46.     The  wire  should 


wire 


current 


wire 


side  view 
(wire  seen  endwise) 


top  view 


Fig.  46. 


be  covered  with  white  cotton  or  silk  so  as  to  be  easily  visible. 
Connect  the  wire  through  a  suitable  rheostat  and  through  a 
reversing  switch  to  no-volt,  direct-current  supply  mains,  and 


ELECTRICITY  AND   MAGNETISM.  113 

show  that  the  wire  is  pushed  sidewise  as  indicated  by  the  dotted 
arrow  in  the  side  view.  Reverse  the  current  and  show  that 
side  force  is  reversed. 

It  is  remarkable  how  widely  prevalent  is  the  idea  of  attraction; 
point  out  explicitly  that  the  wire  is  not  attracted  by  the  pole  of 
the  magnet  but  pushed  sidewise  as  indicated. 

(b)  Project  the  image  of  a  direct-current  arc  (using  ordinary 
arc  lamp  carbons)  on  the  screen.     Bring  the  north  pole  of  a  bar 
magnet  up  behind  the  arc  and  call  attention  to  sidewise  movement 
of  image  of  arc  on  screen.     Reverse  the  magnet  and  call  attention 
to  reversed  sidewise  movement  of  image.     Bring  magnet  nearer 
and  nearer  to  arc  until  arc  is  blown  out. 

(c)  Arrange  a  single-pole  single-throw  switch  in  a  circuit  in 
which  10  or  15  amperes  of  current  flows  from  no-volt,  direct- 
current  supply  mains.     Open  the  switch  slowly  and  call  attention 
to  the  destructive  arc  which  is  produced.     Show  how  greatly 
this  arc  is  reduced  by  opening  the  switch  quickly.     Place  a 
magnet  as  indicated  in  Fig.  140,  page  203,  General  Physics,  and 
show   that  arc  is  almost    instantly  blown   out  if    the  switch 
is  opened  slowly  or  rapidly. 

83.  The  electric  motor. — Exhibit  a  commercial  type  of  direct- 
current  motor,  and  take  out  the  armature  so  that  the  armature 
wires  may  be  seen  lying  on  the  armature  surface  parallel  to  the 
armature  shaft. 

The  small,  non-enclosed,  two-pole,  ring-wound,  Crocker- 
Wheeler  motor  is  one  of  the  best  motors  for  this  experiment. 
Excite  the  field  magnet  of  the  motor  by  connecting  the  field 
winding  through  a  controlling  switch  to  no- volt,  direct-current 
supply  mains;  connect  the  brushes  through  a  suitable  rheostat 
and  a  controlling  switch  to  no- volt  direct-current  mains;  and 
operate  the  motor.  Show  that  direction  of  running  is  reversed 
by  reversing  field  connections,  or  armature  connections;  but 
that  reversal  of  both  leaves  direction  of  running  unchanged. 

If  a  motor  having  a  drum- wound  armature  is  used,  the  student 
9 


114  CALENDAR  OF  LEADING   EXPERIMENTS. 

should  be  required  to  read  the  fine  print  on  page  209,  General 
Physics. 

84.  The  chemical  effect  of  the  electric  current  may  be  strik- 
ingly shown  by  the  decomposition  of  lead  nitrate  as  explained  in 
Art.  147,  page  214,  General  Physics.     The  best  arrangement  is  to 
use  a  narrow  cell  with  plate  glass  sides  and  place  it  in  the  lantern 
(horizontal  projection  arrangement) .     After  the  deposition  of  lead 
crystals  on  one  electrode,  reverse  the  battery  connections  when 
these  crystals  will  be  seen  to  redissolve,  and  crystals  will  grow 
on  the  other  electrode. 

Where  lead  crystals  are  being  deposited  the  density  of  the 
solution  is  reduced  as  may  be  seen  by  the  upward  streaming  of 
the  solution  near  the  cathode,  and  where  the  previously  deposited 
crystals  are  being  redissolved  the  density  of  the  solution  is 
increased  as  may  be  seen  by  the  downward  streaming  of  the 
solution  near  the  anode.  Point  out  that  the  chemical  action 
in  an  electrolytic  cell  takes  place  only  at  the  electrodes. 

The  voltaic  cell. — It  is  very  important  to  direct  the  students' 
attention  very  particularly  to  Art.  149,  page  216,  General  Physics. 
A  voltaic  cell  is  any  electrolytic  cell  in  which  the  chemical  action 
produced  by  the  current  is  a  source  of  energy. 

85.  The  simple  voltaic  cell.    Voltaic  action  and  local  action. — 

Place  the  above  described  glass  cell  in  the  lantern  (horizontal  pro- 
jection arrangement),  fill  the  cell  with  dilute  sulphuric  acid,  and 
introduce  a  strip  of  clean  pure  zinc.  No  perceptible  chemical 
action  takes  place,  that  is  to  say,  no  bubbles  of  hydrogen  are 
seen  to  be  produced. 

Connect  the  strip  of  zinc  to  a  strip  of  copper  by  a  soldered 
wire  connection  and  place  the  two  strips  into  the  cell.  An 
electric  current  now  flows  through  the  wire  from  copper  to  zinc 
and  through  the  acid  from  zinc  to  copper,  this  current  decom- 
poses the  H2SO4,  the  SC>4  radical  is  liberated  at  the  zinc  electrode 
where  it  combines  with  the  zinc  to  form  ZnSO4,  and  the  hydrogen 
is  liberated  at  the  copper  strip  where  it  appears  in  the  form  of 


ELECTRICITY  AND   MAGNETISM.  115 

bubbles.  Cut  the  wire  and  this  action  ceases.  This  is  voltaic 
action. 

Dip  the  clean  strip  of  zinc  momentarily  into  a  dilute  solution 
of  copper  sulphate,  and  minute  particles  of  copper  will  be  left 
on  the  zinc.  Replace  the  contaminated  strip  of  zinc  in  the 
dilute  sulphuric  acid,  and  chemical  action  takes  place  as  indicated 
by  the  formation  of  hydrogen  bubbles.  This  is  local  action. 

Connect  the  strip  of  contaminated  zinc  to  a  strip  of  copper 
by  a  soldered  wire  connection  and  place  both  strips  in  the 
sulphuric  acid.  Hydrogen  bubbles  will  be  seen  to  form  at  both 
electrodes,  that  is  to  say,  local  action  and  voltaic  action  both 
take  place. 

Amalgamate  the  contaminated  strip  of  zinc  by  rubbing  a 
little  mercury  over  it,  replace  the  two  strips  in  the  sulphuric 
acid  cell,  and  voltaic  action,  only,  will  be  in  evidence  by  the 
liberation  of  hydrogen  bubbles  at  the  copper  strip.  Cut  the 
wire  and  this  voltaic  action  will  cease. 

86.  Experiment  showing  the  use  of  oxidizing  agent  around 
the  cathode  (copper  or  carbon)  of  a  voltaic  cell. — This  matter 
is  explained  in  Art.  153,  page  220,  General  Physics,  and  it  may 
be  shown  experimentally  as  follows:    Connect  an  ammeter  to 
a  simple  voltaic  cell,  and  allow  the  current  to  flow  until  the 
ammeter  reading  falls  considerably  (until  the  dissolved  atmo- 
spheric oxygen  near  the  copper  electrode  is  used  up  and  hydrogen 
bubbles  are  given  off  copiously).     Then  pour  into  the  cell  a 
solution   of   potassium   bichromate.     The   ammeter   reading   is 
thereby  increased  and  hydrogen  bubbles  are  no  longer  liberated. 

87.  Change  of  resistance  with  temperature. — (a)  A  battery 
jar  is  provided  with  two  lead  electrodes.     Fill  the  jar  nearly 
full  of  water,  connect  through  an  ammeter  (reading  up  to  20  or 
30  amperes)  to  no- volt  direct  current  supply  mains,  and  connect 
a  voltmeter  between  the  electrodes  (no- volt  meter).     Place  a 
thermometer  in  the  jar,  and  pour  dilute  sulphuric  acid  into  the 
jar  very  slowly  with  vigorous  stirring  until  the  ammeter  reading 


n6 


CALENDAR   OF   LEADING  EXPERIMENTS. 


is  about  5  amperes.  Then  read  thermometer,  voltmeter  and 
ammeter. 

Allow  the  current  to  flow  for  a  while  until  the  temperature 
rises  considerably.  Then  again  take  the  readings  of  thermom- 
eter, voltmeter  and  ammeter. 

Ignoring  the  small  polarization  voltage  (2  or  3  volts)  the 
resistance  of  the  acid  solution  at  each  temperature  can  be  found 
from  the  corresponding  readings  of  voltmeter  and  ammeter. 

(b)  File  off  the  corners  of  three  4"  X  8"  pieces  of  window 
glass.  Wind  about  100  feet  of  No.  28  B.  &  S.  cotton-covered 
copper  wire  in  one  layer  on  each  strip,  leaving  3  inches  of  one 
end  of  each  piece  of  glass  uncovered.  Clamp  the  bare  ends 
of  the  glass  strips  to  a  wooden  block,  connect  all  of  the  windings 
of  wire  in  series  and  bring  the  terminals  to  two  binding  posts 
on  the  wooden  block.  Place  the  wooden  block  across  the  top 
of  a  battery  jar  with  the  wire-covered  strips  of  glass  projecting 
into  the  jar.  Fill  the  jar  with  kerosene  so  as  to  cover  the  wire 
windings,  bore  a  hole  through  the  wooden  cover  for  a  ther- 
mometer, and  arrange  a  good  stirrer. 

Connect  as  under  a  above  to  no- volt  direct-current  supply 
mains,  and  determine  the  resistance  of  the  copper  wire  at,  say, 
25°  C.  and  at  75°  C. 

88.  Conductivity  of  glass  at  high  temperatures. — A  thin- 
walled  glass  tube  about  one  inch  in  diameter  and  8  or  10  inches 
long  is  connected  as  shown  in  Fig.  47,  and  one  side  of  the  tube  is 


supply 
mains 


-T~">-'^-%/W\/VvJ 3 

rheostat       g 


o            t 

0        1 

glass 

gjloo  volts 

tube 

0                1 

Fig.  47. 

heated  by  sweeping  the  flame  of  a  blast  lamp  up  and  down 
between  the  wire  terminals.  Keep  the  supply  switch  open  until 
all  connections  ate  made,  and  use  great  caution  to  avoid  contact 


ELECTRICITY  AND   MAGNETISM.  117 

with  high-voltage  circuit  after  the  supply  switch  is  closed.  The 
current  starts  along  a  very  narrow  path,  quickly  heats  the  narrow 
strip  to  a  very  high  temperature,  and  the  resistance  of  this  path 
of  very  hot  glass  drops  to  200  ohms  or  less.  The  lo-ohm,  10- 
ampere  rheostat  is  necessary  to  prevent  an  excessive  flow  of 
current  after  the  glass  begins  to  conduct  appreciably. 

Electricity  or  energy;  which? — Article  170,  page  238,  General 
Physics,  is  intended  to  turn  the  student  away  from  any  premature 
incursion  into  the  atomic  theory  of  electricity,  and  hold  his 
attention  fast  to  the  most  important  practical  phase  of  electrical 
science,  namely,  the  purely  mechanical  phase.  It  is  conceivable 
that  the  atomic  conceptions  of  electrical  phenomena  may  some 
time  come  to  be  important  in  everyday  life  and  in  everyday 
engineering,  but  that  time  is  certainly  not  yet;  although  the 
atomic  theory  is  nearly  as  important  in  engineering  research  as 
in  any  other  branch  of  physical  research. 

89.  Experiment  with  fan  blower  and  dynamo  generator. — 
Drive  a  fan  blower  by  a  Drive  a  small  shunt  dynamo 
small,  direct-current,  shunt  as  a  generator  by  means  of  a 
motor  with  an  ammeter  in  the  small,  direct-current,  shunt 
armature  circuit  of  the  motor  motor  with  an  ammeter  in  the 
to  indicate  roughly  the  power  armature  circuit  of  the  motor 
delivered  to  the  blower.  to  indicate  roughly  the  power 

required  to  drive  the  generator. 

Arrange  a  gate  so  that  the  Arrange  a  rheostat  and  switch 
outlet  of  the  blower  can  be  so  that  the  generator  can  be 
closed  or  opened  to  any  desired  operated  on  open  circuit  (arma- 
extent.  ture  delivering  no  current)  or 

so  that  the  current  delivery  can 
be  varied  at  will. 

Arrange  an  open-tube  manom-  Connect  a  voltmeter  across 
eter  to  measure  the  air  pres-  the  armature  terminals  of  the 
sure  behind  the  gate.  generator. 


Il8  CALENDAR  OF  LEADING  EXPERIMENTS. 

With  gate  closed  note  power  With  armature  circuit  of 
required  to  drive  the  fan  and  generator  open  note  power 
note  value  of  air  pressure  E.  required  to  drive  the  generator 

and  note  voltage  E  across 
terminals  of  generator  arma- 
ture. 

Opening  gate  wider  and  Closing  switch  and  adjusting 
wider,  note  increase  of  power  rheostat  so  as  to  take  more  and 
required  to  drive  the  fan  and  more  current  from  generator 
note  slight  decrease  of  pressure  armature,  note  increase  of 
E  behind  gate.  power  required  to  drive  gen- 

erator and  note  slight  decrease 
of  voltage  E  between  armature 
terminals. 

An  interesting  calculation  is  that  which  is  suggested  in  problem 
198,  page  568,  General  Physics,  and  approximate  data  for  such  a 
calculation  can  be  obtained  from  the  motor-generator  arrange- 
ment here  described  if  an  ammeter  is  placed  in  the  armature  circuit 
of  the  generator.  Thus  if  5  amperes  of  current  is  taken  by  the 
motor  armature  from  no-volt  direct-current  supply  mains  when 
generator  is  delivering  no  current,  and  if  15  amperes  is  taken  by 
motor  armature  when  the  generator  is  delivering  9  amperes. 
Then,  neglecting  RI2  losses  in  motor  armature  and  in  dynamo 
armature,  we  may  take  (15  amperes  —  5  amperes)  X  no  volts  as 
the  part  of  the  power  which  is  supplied  to  the  generator  by  the 
motor  and  delivered  as  output  by  the  generator  so  that  the 
voltage  E  of  the  generator  must  be  approximately  such  as  to 
give 

9  amperes  X  E  =  (15  amperes  —  5  amperes)  X  no  volts. 

90.  Experiment  on  back  electromotive  force  of  a  motor. — 

Connect  the  field  winding  F  of  a  direct-current  shunt  motor 
directly  to  no- volt  direct-current  supply  mains  through  a  con- 
trolling switch  5;  connect  the  armature  M  of  the  motor  to  the 


ELECTRICITY  AND   MAGNETISM.  119 

same  mains  through  a  rheostat,  an  ammeter  A  and  a  con- 
trolling switch  S';  and  connect  a  voltmeter  V  to  terminals 
of  M  as  shown  in  Fig.  48. 


Fig.  48. 

To  determine  resistance  of  M  lock  the  armature  so  that  it 
cannot  rotate,  and  reduce  resistance  in  the  rheostat  until  the  rated 
full-load  current  flows  through  A  and  M ;  then  take  readings  of 
A  and  V.  The  required  resistance  of  M  is  equal  to  V/A  accord- 
ing to  Ohm's  law. 

Then  unlock  the  motor  armature  and  allow  it  to  run,  without 
a  belt  load,  let  us  say,  and  note  the  great  decrease  of  current 
through  A  and  M  and  the  great  increase  of  voltage  indicated 
by  V.  Take  readings  A'  and  V  when  motor  is  running  at 
full  speed. 

We  thus  get  data  for  a  calculation  like  that  of  problem  215, 
page  571,  General  Physics. 

Note. — The  important  thing  in  this  experiment  is  to  show  the 
great  decrease  of  current  through  M  when  the  motor  is  allowed 
to  start  and  rise  to  full  speed. 

91.  The  transformer  experiment. — A  laminated  iron  core  about 
i"  X  i"  X  8"  long  has  about  150  turns  of  No.  16,  B.  &  S. 
copper  wire  wound  upon  it,  the  wound  core  is  mounted  in  an 
upright  position  on  a  wooden  base,  and  the  terminals  of  the 
winding  are  brought  out  to  binding  posts  as  indicated  in  Fig. 
198,  page  266,  General  Physics.  This  winding  can  be  safely 
connected  directly  to  no- volt,  6o-cycle,  alternating-current 
supply  mains. 


120  CALENDAR  OF  LEADING   EXPERIMENTS. 

(a)  A  copper  ring  2  or  3  inches  in  diameter,  J  inch  wide  and 
J  inch  thick  is  held  in  a  pair  of  strong  pliers  and  brought  down 
over  the  above  wire-wound  core.  The  repeated  reversals  of 
magnetism  of  the  core  induce  an  alternating  current  in  the  ring 
and  the  ring  becomes  red  hot. 

(6)  Hold  the  cold  ring  loosely  by  the  fingers  so  as  to  encircle 
the  core  about  2  inches  from  its  upper  end,  and  suddenly  close 
the  supply  switch.  The  ring  is  thrown  violently  upwards. 

(c)  A  small  coil  of  insulated  copper  wire  three  or  four  inches  in 
diameter  and  containing  15  or  20  turns  of  wire  has  a  short  piece 
of  German  silver  wire  (No.  22,  B.  &  S.,  let  us  say)  connected 
across  its  terminals,  and  the  coil  is  held  in  the  hand  and  brought 
slowly  down  over  the  above  wire-wound  core.  The  small  coil 
acts  as  the  secondary  coil  of  a  transformer  (a  step-down  trans- 
former), and  an  alternating  current  is  induced  in  it  as  indicated 
by  the  heating  of  the  German  silver  wire. 

92.  Eddy  currents  in  a  solid  iron  rod. — A  coil  of  insulated, 
No.  16,  B.  &  S.,  copper  wire  containing  about  1,000  turns  and 
having  an  opening  about  3  inches  in  diameter  is  connected  to 
no- volt,  6o-cycle,  alternating  current-supply  mains  through  a 
suitable  rheostat  (seven  or  eight  ohms  and  ten-ampere  capacity) 
and  a  controlling  rheostat.     A  short  iron  rod  2  or  2\  inches  in 
diameter  with  a  strong  handle  like  a  soldering  iron  is  thrust  into 
the  opening  of  the  coil,  and  the  rod  soon  becomes  very  hot 
because  of  eddy  currents. 

Eddy  currents  due  to  motion  of  a  good  electrical  conductor 
near  an  unvarying  magnet  may  be  shown  by  dropping  a  thick 
sheet  of  copper  into  the  space  between  the  two  facing  poles  of  a 
strong  electromagnet. 

93.  Demagnetization  by  a  weakening  series  of  reversals. — 
A  steel  magnet  placed  in  the  coil  which  is  described  in  experiment 
92  and  slowly  withdrawn  is  left  almost  wholly  demagnetized. 
A  magnet  (or  a  magnetized  watch)  may  be  demagnetized  by 
holding  it  near  the  pole  of  a  strong  magnet  and  turning  it  slowly 


ELECTRICITY  AND   MAGNETISM.  121 

while  moving  it  farther  and  farther  away  from  the  magnet  pole. 
The  axis  of  turning  must  be  at  each  place  at  right  angles  to  the 
lines  of  force  of  the  magnetic  field  at  that  place. 

94.  Spark  at  break.  —  Connect  the  coil  and  core  which  are 
described  in  Experiment  96  to  two  cells  of  storage  battery  in 
series,  and  arrange  a  strip  of  spring  brass   BB   as  shown  in  Fig. 
49  for  quickly  breaking  the  cir- 
cuit at     p     (see    also  Fig.  215, 


page  276,  General  Physics)  .    Also.      ^  block 

arrange  a  piece  of  German  sil- 
ver wire  of  the  same  resistance 

as  the  coil  to  be  put  in  circuit  in  place  of  the  coil  and  permit 
the  same  steady  current  to  flow.  A  small  glow  lamp  may  be 
included  in  the  circuit  to  show  by  its  brightness  that  the  cur- 
rent has  approximately  the  same  value  in  each  part  a,  b  and 
c  of  the  experiment,  as  follows: 

(a)  With  German  silver  wire  in  place  of  the  coil,  break  the 
circuit  suddenly  at     p     and  note  that  there  is  no  perceptible 
spark  at  break. 

(b)  Place  the  coil  in  circuit  without  the  laminated  iron  core, 
and  show  that  there  is  a  moderate  spark  at  break 

(c)  Place  the  laminated  core  in  the  coil  and  show  that  there 
is  a  very  intense  spark  at  break. 

An  interesting  experiment  for  the  individual  student  is  to 
connect  a  telegraph  relay  as  a  buzzer  (see  problem  General 
Physics),  operate  it  by  three  or  four  dry  cells,  and  connect  hand- 
electrodes  to  the  terminals  of  the  magnet  winding.  At  each 
break  of  the  circuit  a  moderately  high  voltage  exists  across 
the  magnet  winding  which  may  be  felt  by  taking  hold  of  the 
hand  electrodes. 

95.  Choking  effect  of  an  inductance  in  an  alternating-current 
circuit.  —  Connect  a  6owatt,  no-  volt  tungsten  lamp  to  no-volt, 
6o-cycle,  alternating-current  supply  mains  in  series  with  the 
German  silver  resistance  referred  to  in  the  previous  experiment. 


122  CALENDAR  OF  LEADING  EXPERIMENTS. 

Connect  a  similar  lamp  to  the  same  supply  mains  in  series  with 
the  coil  referred  to  in  the  previous  experiment.  The  resistances 
of  the  two  circuits  are  the  same,  but  the  lamp  which  is  in  series 
with  the  coil  is  dimmer  than  the  other  lamp  (showing  that  less 
alternating  current  flows  through  it),  and  if  the  laminated  iron 
core  is  placed  in  the  coil  the  current  will  be  very  greatly  reduced. 
See  page  279,  General  Physics. 

Note. — Change-over  switches  may  be  arranged  to  shift  the 
above  circuits  from  no-volt,  alternating-current  supply  mains 
to  no-volt,  direct-current  supply  mains. 

96.  Experiment  showing  slow  growth  of  current  in  an  inductive 
circuit. — A  small  glow  lamp  is  connected  to  a  battery  through  a 
controlling  switch  and  in  series  with  a  coil  of  wire  in  which  is  a 
removable  iron  core.  When  the  switch  is  closed  the  lamp  comes 
almost  instantly  to  full  brightness  when  the  iron  core  is  removed, 
but  it  takes  a  very  perceptible  time  for  the  lamp  to  come  to  full 
brightness  after  closing  the  switch  when  the  iron  core  is  in  place. 

A  delay  of  about  0.6  second  with  iron  core  in  place  will  be 
obtained  under  following  specifications:  Use  a  half-ampere,  2- 
volt,  tungsten  filament  lamp,  and  two  storage  battery  cells 
connected  in  series.  Make  the  coil  about  6  inches  long  with  i 
inch  depth  of  winding  and  containing  about  1,350  turns  of  No. 
14,  B.  &S.,  double  cotton-covered  copper  wire.  Make  the  core 
1 1  inch  X  1 1  inch  X  13  inches  long,  built  up  of  soft  sheet-iron 
stampings  bound  together  by  one  layer  of  strong  twine.  The 
corners  of  the  core  should  be  rounded  off  to  avoid  cutting  of 
twine.  Soak  the  whole  core  in  shellac  varnish  and  bake.  Each 
end  of  the  core  for  about  3^  inches  should  be  bare  so  that  the 
edges  of  the  laminations  of  the  core  can  come  as  close  as  possible 
to  the  edges  of  laminations  of  a  yoke.  Make  area  of  contact 
of  each  end  of  core  with  yoke  about  ij  X  3  inches.  The  coil 
must  have  a  2|-inch  hole  to  admit  the  laminated  core.  The 
yoke  should  be  mounted  on  a  base  board  with  lugs  projecting 
upwards  to  connect  with  and  support  the  ends  of  the  core. 


ELECTRICITY  AND   MAGNETISM.  123 

A  very  interesting  experiment  with  the  above  apparatus  is 
the  following:  Connect  lamp  and  coil  in  series  to  the  2  cells  of 
storage  battery,  placing  the  coil  on  its  side  on  the  table  (yoke 
not  used).  Allow  the  magnetic  forces  to  draw  the  core  quickly 
into  the  coil  and  the  lamp  will  be  momentarily  dimmed.  Draw 
the  core  quickly  out  of  the  coil  and  the  lamp  will  become  momen- 
tarily very  brilliant.  (Be  careful  not  to  burn  out  the  lamp  by 
withdrawing  core  too  quickly.) 

While  the  core  is  being  drawn  into  the  coil  the  power,  El, 
delivered  to  the  coil  by  the  battery  is  being  only  in  part  used  to 
heat  the  wire  of  the  coil  (RIZ)  because  some  power  is  evidently 
used  to  pull  the  core  into  the  coil.  Therefore  RP  is  less  than 
El,  or  RI  is  less  than  E,  or  J  is  less  than  E/R,  which  is 
the  value  of  the  steady  current  according  to  Ohm's  law.  While 
the  core  is  being  drawn  into  the  coil  the  magnetic  flux  through 
the  coil  is  increasing,  this  increasing  flux  induces  a  back  electro- 
motive force  in  the  coil  (opposing  the  current),  and  the  work 
done  in  forcing  the  current  to  flow  in  opposition  to  this  back 
electromotive  force  is  spent  in  two  ways,  namely,  in  pulling  the 
core  into  the  coil  and  in  magnetizing  the  core. 

Inductance  of  a  coil  or  circuit. — The  discussion  of  inductance 
as  given  in  General  Physics  is  seriously  incomplete.  Indeed  we 
have  never  written  a  book  without  being  unduly  influenced  by 
a  feeling  of  the  hopelessness  of  the  undertaking  of  bringing  young 
men  to  a  fairly  complete  understanding  of  the  fundamentals  of 
physics,  and  we  therefore  cut  out  things  that  should  be  included. 
We  might  excuse  such  omissions  by  saying  that  we  do  not  wish 
to  make  a  perfect  book  and  thereby  deprive  all  physics  teachers 
of  their  work! 

A  coil,  having  Z  turns  of  wire,  has  a  current  i  flowing  in  it, 
as  indicated  in  Fig.  50.  The  current  in  the  coil  produces  a 
magnetic  field  in  the  surrounding  region  as  indicated  by  the  lines 
of  force  in  Fig.  50,  and  it  can  be  shown*  from  the  magnetic 
definition  of  current  strength  (see  Art.  144,  page  209,  General 

*  It  is  not  worth  while  to  introduce  this  proof  here. 


124  CALENDAR  OF   LEADING   EXPERIMENTS. 

Physics)  that  the  intensity  of  the  magnetic  field  is  everywhere 
doubled  in  value  if   i   is  doubled  in  value,  the  trend  of  the  lines 

of  force  remaining  unchanged.  There- 
of wire  fore  the  amount  of  magnetic  flux 
through  each  turn  of  wire  or  the  av- 
erage flux  per  turn,  <£,  is  doubled  if 
i  is  doubled.  Therefore  $  is  pro- 
portional  to  i  so  that  we  may  write 

Fig.  SO.  $>  =  bi  (i) 

where   b   is  a  constant  for  a  given  coil  or  circuit.     If   i   changes 
it  is  evident  that    $   must  change   b   times  as  fast,  or 

d*         di 


But  the  changing     $     induces  an  electromotive  force      —  — 

d$ 

in  each  turn  of  wire  or  a  total  electromotive  force    —  Z  -3-    in 

at 

the  whole  coil,  according  to  equation  (82),  page  257,  General 

d^&  fi^i 

Physics,  or  using  the  value  of     -j-     from  (ii)  we  get    —  bZ  — 

at  at 

as  an  expression  for  the  electromotive  force  induced  in  the  coil 
by  the  changing  current,  and,  if  we  represent  the  quantity  bZ 
by  the  single  letter  L,  we  have 


where   Ef    is  the  electromotive  force  induced  in  a  coil  or  circuit 
by  the  changing  of  the  current.     The  negative  sign  indicates 

di 
that    Ef    is  opposed  to    i   when    —    is  positive,  that  is,  when    i 

is  increasing.     Therefore,  to  make  the  current  increase  at  the 

di 

rate   —  ,   an  outside  electromotive  force  (the  electromotive  force 
at 

of  a  battery,  for  example)  equal  to    L—     but  helping  the  current 


ELECTRICITY  AND   MAGNETISM.  125 

must  act  or  push  on  the  circuit.     That  is 

;'.       B-L$      ..         V:       .  (iv) 

which  is  equation  (84),  page  278,  General  Physics,  and  in  which 
E     is  the  outside  electromotive  force  required  to  make  the 

current  in  a  coil  or  circuit  increase  at  the  rate    —  ,    and   L    is  a 

CLt 

constant  for  the  given  circuit.     The  quantity    L    is  called  the 
inductance  of  the  circuit. 

Let  R  be  the  resistance  of  the  circuit,  and  let  E  be  the  total 
electromotive  force  acting  on  the  circuit.  Then  a  portion,  Ri, 
of  the  electromotive  force  is  used  to  overcome  resistance,  and  the 
remainder,  E  —  Ri,  causes  the  current  to  increase  in  accordance 
with  equation  (iv). 

Note. — If  there  is  an  iron  core  in  the  coil  in  Fig.  50  the  flux  4? 
will  be  nearly  proportional  to  i  for  small  values  of  i,  but  as  the 
iron  approaches  magnetic  saturation  the  flux  $  is  no  longer 
even  approximately  proportional  to  i. 

Flux-turns. — The  quantity  3>  in  the  above  discussion  is  the 
average  flux  per  turn  of  wire,  and  the  product  $Z  might  properly 
be  called  the  total  flux  passing  through  the  coil  (each  part  of  the 
actual  flux  in  Fig.  50  being  counted  n  times,  where  n  is  the 
number  of  turns  of  wire  surrounding  that  part).  This  product 
$Z  is  usually  called  the  number  of  linkages  of  lines  of  force  and 
turns  of  wire,  and  it  is  expressed  as  flux-turns.  Multiply  both 
members  of  equation  (i)  by  Z  and  we  get  3>Z  =  bZi.  But  bZ 
is  the  inductance  L  of  the  coil,  so  that 

$Z  =  Li.  (v) 

Consider  a  canal  boat  on  Consider  a  circuit  on  which 
which  a  propelling  force  E  is  an  electromotive  force  E  is  act- 
exerted  by  the  tow  rope.  As-  ing.  The  portion  of  E  which 
sume  the  backward  drag  of  the  is  used  to  overcome  friction  is 
water  to  be  proportional  to  the  Ri,  when  R  is  a  constant  and 


126 


CALENDAR  OF   LEADING   EXPERIMENTS. 


velocity  i  of  the  boat.     Then  i   is  the  current,  and  the   net 

the    net  accelerating   force  is  accelerating  electromotive  force 

E—  Ri,  where  R  is  a  constant,  is  E  —  Ri,  so  that 
and  we  have 


E  -  Ri  =  L 


di 
dt 


E  -  Ri  =  L 


(vi) 


di 

dt 


where   L  is  the  mass  of    the 

di 

boat  and  -T.  is  its  acceleration. 
at 

When  the  boat  reaches  full 


where   L   is  the  inductance  of 

di 

the  circuit  and  -j-.  is  the  rate  of 
dt 

increase  of  the  current. 

When  the  current  reaches  full 


speed  (corresponding  to  given     value   (corresponding  to  given 


force    E)     then     -r  = 
aquation  (vi)  becomes 


o     and 


di 

value  of  E)  then  -r  = 
at 

equation  (vii)  becomes 


o    and 


•p 

E  =  Ri     or     i=  —    (viii) 
J\. 


E  =  Ri     or     i  =  —       (ix) 
K. 


At  the  instant  that  the  mule         At  the  instant  that  the  elec- 
begins  to  pull,  i=  o  andequa-     tromotive  force  E  begins  to  act 


tion  (vi)  becomes 


(x) 


on  the  circuit,  i  =  o  and  equa- 
tion (vii)  becomes 

E  =  L-  (xi) 


When  the  mule  stops  pulling  When  the  electromotive  force 

after  having  set    the    boat  in  E   ceases  to  act   after  having 

motion,    E  =  o   and   equation  established  a    current,    E  =  O 

(vi)  becomes  and  equation  (vii)  becomes 

di  rdi 

L T;  =  —  la        (xn)  L  —  =  -  Ri           (xm) 


Note. — Equation  (vi)  is  usually  written  and  thought  of  in  the 

di 

form  E  =  Ri  +  L  —  ,  that  is  to  say,  the  total  electromotive  force 
dt 

acting  on  a  simple  circuit  of  resistance  R  and  inductance  L  is 
partly  used  to  "  overcome  "  resistance  and  partly  used  to  "  over- 
come "  inductance;  the  former  part  is  Ri  and  the  latter  part 


s 


ELECTRICITY  AND   MAGNETISM.  127 

97.  Elimination  of  spark  at  break  by  a  condenser. — Figure  51 
shows  the  coil  L  and  the  quick-break  arrangement  p  (as 
described  in  experiment  94)  connected  to  2  storage  battery  cells 


Fig.  51. 

B.  As  stated  in  experiment  94  an  intense  spark  at  break  is 
produced  when  the  iron  core  is  in  the  coil,  but  when  a  condenser 
C  is  connected  as  shown  the  spark  at  break  is  eliminated.  The 
condenser  should  have  about  one  microfarad  capacity. 

The  reversed  surge  of  current  which  flows  through  the  coil 
L  and  the  battery  B  after  breaking  the  circuit  at  p  may  be 
shown  by  its  demagnetizing  action  on  the  iron  core  as  explained 
on  page  283,  General  Physics. 

The  so-called  discharge  resistance. — Connect  a  glow  lamp 
in  place  of  the  condenser  in  Fig.  51  and  show  that  spark  at  break 
is  eliminated.     It  is  common  practice  to  use  a  resistance  thus 
connected  across  a  switch  which  is 
used  to  open  a  highly  inductive  cir- 
cuit, such  as  the  field  winding  of 
>  a  large  dynamo,  and  the  switch  is 
always    arranged    as    a    two-step  Fig  52> 

switch.     The  first  step  inserts  the 

resistance  in  circuit  and  the  second  step  opens  the  circuit  after 
the  current  has  been  greatly  reduced  by  the  insertion  of  the  re- 
sistance. A  common  arrangement  of  switch  with  ''discharge 
resistance"  is  shown  in  Fig.  52. 

98.  Flow  of  alternating  current  through  a  circuit  containing  a 
condenser. — Connect  a  25-watt,  no- volt,  tungsten  filament, 
glow  lamp  in  series  with  a  10-  or  2O-microfarad  condenser,  and 
arrange  a  change-over  switch  so  that  the  condenser  and  lamp 


128 


CALENDAR  OF   LEADING   EXPERIMENTS. 


can  be  connected  to  no-volt,  direct-current,  supply  mains  or  to 
6ocycle,  no- volt,  alternating-current,  supply  mains  as  indicated 
in  Figs.  222  and  223,  page  284,  General  Physics. 

99.  The  lightning  arrester. — A  very  interesting  experiment  is 
to  show  a  simple  type  of  lightning  arrester  in  action.  Thus 
Fig.  53  shows  a  single-pole  direct-current  arrester  of  the  magnetic 


supply  mains 


ground 


ifr 


spark 


Fig.  53. 


blow-out  type.  The  ground  connections  in  the  figure  should  be 
good  metallic  connections  to  the  same  piece  of  gas  or  water  pipe 
because  the  action  of  the  magnetic  blow-out  is  most  interesting 
when  the  spark  across  gap  g  produces  a  dead  short  circuit. 
If  22O-volt  direct-current  supply  is  available  it  should  be  used 
in  preference  to  no- volt  supply. 

Connect  one  terminal  of  the  electric  machine  or  induction  coil 
to  ground  and  the  other  terminal  to  the  metal  ball  B.  When  the 
spark  jumps  from  B  to  W  it  is  choked  by  the  winding  L,  and 

s  E  therefore  the  sudden  spark  discharge 

M 03 ,      0 

}'  I   passes  across  gap    g   to  ground.     But 

the  spark  across  g  starts  an  electric 
arc  and  introduces  a  dead  short  cir- 
cuit between  the  supply  mains.  The 
blow-out  magnet  L  is  to  eliminate 
this  short  circuit. 


Fig.  54. 

brass  strip     B     (faced 
short-circuit   when    G 


100.   Electrostatic    attraction.  —  A 

with  varnished    paper    so   as    to  avoid 
touches  it),  and  a   suspended  strip  of 


ELECTRICITY  AND   MAGNETISM. 


129 


gold  leaf  G  as  indicated  in  Fig.  54  are  placed  in  the  lantern 
(horizontal  projection  arrangement),  and  the  binding  posts  are 
connected  to  no- volt,  direct-current  supply  mains. 

Show  that  the  attraction  is  sensibly  the  same  if  the  binding 
posts  are  connected  to  I  lO-volt,  alternating-current  supply  mains. 

Exhibit  an  electrostatic  voltmeter. 

10 1.  Capacity  of  a  condenser. — The  experiment  which  is  de- 
scribed in  Art.  204,  page  289,  General  Physics,  is  very  helpful  in 
giving  to  the  student  a  clear  understanding  of  equation  (89), 
page  290,  General  Physics. 

The  term  capacity  is  misleading  in  that  it  suggests  that  a  con- 
denser can  hold  only  a  certain  amount  of  charge  just  as  a  pail 
can  hold  only  a  certain  amount  of  water. 

102.  Residual   charge    on   a   condenser.     So-called    electric 
absorption. — 

A  rubber  tube  is  stretched  by  A  Leyden  jar  condenser  is 
a  very  considerable  steady  force  charged  by  a  very  high  steady 
E  and  certain  elongation  Q  is  voltage  R  and  certain  charge 
produced.  Q  is  drawn  out  of  one  coating 

and  pushed  into  the  other  coat- 
ing thus  producing  what  may 
be  thought  of  as  an  electrical 
stress  in  the  glass  wall. 

The  two  coatings  of  the 
Leyden  jar  are  connected  by 
a  conductor  so  that  current  can 
flow,  and  the  charge  on  the 
coatings  together  with  the  elec- 
trical stress  in  the  glass  wall 
disappears. 

The  wire  connection  is  then 
taken  away  so  that  current 
cannot  flow. 


The  end  of  the  rubber  tube 
is  momentarily  released  so  as 
to  be  free  to  move  and  the 
stretch  of  the  tube  is  relieved 
by  the  movement. 


The  free  end  of  the  momen- 
tarily relieved  rubber  tube  is 
then  made  fast  so  that  it  cannot 


move. 
10 


130 


CALENDAR  OF   LEADING  EXPERIMENTS. 


A  minute  later  the  end  of  the 
tube  is  released  and  the  move- 
ment of  the  end  shows  that  a 
small  amount  of  stretched  condi- 
tion has  developed  in  the  tube. 

When  a  stretched  rubber 
band  is  momentarily  released 
most  of  the  stretch  disappears 
at  once,  but  a  small  residue  of 
stretch  disappears  very  slowly; 
and  this  residue  of  stretch  ac- 
cumulates as  sensible  stretch 
if  the  momentarily  released  end 
of  the  rubber  band  is  quickly 
made  fast. 


A  minute  later  the  two  coat- 
ings are  connected  by  wire  and 
a  minute  spark  shows  that  a 
small  amount  of  charged  condi- 
tion has  developed. 

When  the  plates  of  a  charged 
condenser  are  momentarily 
connected  by  a  wire,  most  of 
the  electrical  stress  in  the  di- 
electric disappears  at  once,  but 
a  small  residue  of  the  stress 
disappears  very  slowly;  and 
this  residue  of  stress  accumu- 
lates as  sensible  charge  on  the 
condenser  plates  if  the  momen- 
tary wire  connection  is  taken 
away  so  that  no  current  can 
flow. 

103.  The  use  of  the  spark  gauge. — The  simple  experiment 
which  is  described  in  Art.  210,  pages  298-299,  General  Physics, 
is  best  carried  out  by  the  student  in  the  laboratory. 

104.  The  ringing  spark. — The  oscillatory  character  of  the  dis- 
charge of  a  condenser  through  an  inductance  can  be  shown  by 


O 

to 
high  voltage 
supply 

~~                            o 

£ 

o 
o 

b 

spark  gap 
*  »  J 

Fig.  55. 


the  arrangement  shown  in  Fig.  55.  The  condenser  C  is  charged 
to  about  20,000  or  30,000  volts  by  a  large  influence  machine, 
and  when  it  discharges  across  the  spark  gap  the  back  and  forth 


ELECTRICITY  AND   MAGNETISM.  131 

surging  of  the  discharge  produces  15  or  20  sparks  in  rapid 
succession  and  if  the  frequency  is  low  enough  this  multiple 
spark  gives  a  very  distinct  ringing  sound  like  the  stroke  of  a 
hammer  on  an  anvil.  Also  the  spark  when  reviewed  in  a  rotating 
mirror  is  seen  to  consist  of  a  series  of  sparks  in  rapid  succession. 

The  pitch  of  the  tone  emitted  by  the  spark  can  be  easily 
determined  by  adjusting  a  calibrated  Gal  ton  whistle  to  the 
same  pitch. 

It  is  interesting  to  vary  the  pitch  of  the  spark  by  using  fewer 
Leyden  jars  (smaller  capacity)  or  by  using  a  portion  only  of  the 
sections  of  the  coil  (smaller  inductance).  It  is  advisable  there- 
fore to  bring  out  terminals  from  intermediate  pancake  coils  of 
the  coil  L. 

A  condenser  C,  Fig.  55,  consisting  of  6  fairly  large  Leyden 
jars  grouped  in  parallel  and  a  coil  L  having  about  2,500 
turns  of  double  cotton  covered  No.  18,  B.  &  S.,  copper 
wire  wound  on  a  spool  with  winding  space  8  inches  long  and  I 
inch  deep,  inside  diameter  of  winding  3  inches,  outside  diameter 
of  winding  5  inches,  will  give  a  frequency  of  oscillation  of  about 
2,000  cycles  per  second,  and  at  this  frequency  the  spark  will 
give  a  tone  having  a  pitch  of  4,000  complete  vibrations  per 
second  which  is  suitable  for  this  experiment. 

Note. — Very  high  voltages  come  into  existence  across  the 
terminals  of  L  in  this  experiment  and  therefore  the  coil  should 
be  wound  as  a  set  of,  say,  20  pancake  coils  each  3"  inside 
diameter  X  5  inches  outside  diameter  X  J  inch  thick,  and  these 
pancake  coils  should  be  assembled  on  a  hard  rubber  tube  2 
inches  inside  diameter  and  about  2f  inches  outside  diameter 
with  thin  disks  of  wood  between;  and  the  whole  coil  should  be 
impregnated  with  beeswax  and  rosin  compound. 

105.  The  use  of  the  electric  arc  for  the  maintenance  of  electric 
oscillations. — An  electric  arc  is  maintained  between  a  carbon 
rod  A  and  a  water  cooled  copper  vessel  B,  and  the  arc  is 
shunted  by  a  circuit  containing  an  inductance  L  and  a  con- 


132  CALENDAR  OF  LEADING  EXPERIMENTS. 

denser  C  as  shown  in  Fig.  56.  Under  these  conditions  a 
steady  back  and  forth  surging  of  current  through  the  circuit  LC 
is  maintained. 

Use  as  an  inductance  a  winding  of  about  75  turns  of  No.  16, 
B.  &  S.,  cotton-covered  copper  wire  wound  in  five  or  six  sections 
around  a  short  wooden  cylinder  6  inches  in  diameter,  and  bring 
out  terminals  from  intermediate  sections  so  as  to  be  able  to  use 
15  or  30  or  45  or  60  or  75  turns  at  will.  Use  one  one-microfarad 
condenser  or  two  such  condensers  connected  in  parallel. 


f  000000 

I  L* 

110- volt 

d.  c. 

supply 

carbon 


arc 
copper  vessel 

000000 


Fig.  56. 

With  the  two  condensers  and  with  all  of  the  75  turns  in  cir- 
cuit the  frequency  of  oscillation  will  be  about  4,000  or  5,000 
cycles  per  second  and  the  arc  will  emit  a  shrill  whistling  sound 
of  which  the  pitch  is  8  or  10  thousand  complete  vibrations  per 
second. 

Make  an  open  coil  of  10  or  12  turns  of  No.  18  insulated  copper 
wire,  the  coil  being  large  enough  to  slip  over  the  coil  L  as  above 
described,  connect  a  small  glow  lamp  to  the  terminals  of  this 
coil,  and  bring  the  coil  slowly  over  the  coil  L  while  the  arc  is 
whistling.  Be  careful  not  to  burn  out  the  lamp.  Under  these 
conditions  the  auxiliary  coil  acts  as  the  secondary  of  a  transformer 
as  indicated  by  the  lighting  of  the  lamp. 

Reduce  the  value  of  L  by  using  fewer  sections  as  above 
described,  or  reduce  C,  or  both,  and  the  tone  emitted  by  the 
arc  will  rise  higher  and  higher  in  pitch  and  eventually  pass 
beyond  the  pitch  limit  of  audibility;  but  the  oscillatory  char- 


ELECTRICITY  AND   MAGNETISM.  133 

acter  of  the  current  through  L  and  C  will  still  be  shown  by  the 
lamp.* 

This  experiment  operates  more  satisfactorily  if  the  arc  is  in 
an  atmosphere  of  hydrogen,  but  it  usually  operates  quite  satis- 
factorily with  the  arc  in  air.  A  few  trials  is  sufficient  to  make  it 
go,  especially  if  the  current  through  the  arc  is  reversed  once  or 
twice  to  get  the  more  favorable  direction. 

The  inductances  L^  L%  are  helpful  in  that  they  tend  to  keep 
the  supply  current  at  a  constant  value. 

1 06.  The  electric  field. — The  region  around  a  charged  body 
is  an  electric  field,  and  a  match  stick  suspended  by  a  fine  thread 
may  be  used  for  indicating  the  direction  of  electric  field  in  a 
manner  very  similar  to  the  use  of  the  compass  needle  for  indicat- 
ing the  direction  of  a  magnetic  field.     See  page  305,  General 
Physics. 

107.  Experiment  giving  a  basis  for  the  definition  of  intensity 
of  an  electric  field. — See  pages  306-308,  General  Physics. 

A  very  amusing  experiment  is  to  place  small  pith  figures 
between  two  metal  plates    A  A    and    BB    as  shown  in  Fig.  57, 
the  plates  being  connected  to  the  termi- 
nals of  a  small  influence  electric  machine.     ^ 

The  pith  figures  should  be  made  slightly  j^ 

conducting  by  rubbing  a  trace  of  salt  on     #  B 

them,  and  the  plates  AA  and  BB  should  Fis-  57- 

be  slightly  dished  as  shown. 

1 08.  The  rosin  experiment  is  described  on  page  309,  General 
Physics.     To  get  the  best  results  the  metal  ladle  should  be  con- 
nected to  one  terminal  of  a  large  influence  machine,  and  the 
other  terminal  of  the  machine  should  be  grounded. 

The  gold-leaf  electroscope. — A  very  convenient  arrangement 
of  the  gold-leaf  electroscope  is  shown  in  Fig.  58.  The  leaves  / 

*  The  explanation  of  the  production  of  oscillations  by  the  arc  in  Fig.  56  is  given 
on  pages  141-142  of  W.  S.  Franklin's  Elements  of  Electrical  Engineering,  Vol.  I, 
The  Macmillan  Co.,  1917. 


134 


CALENDAR  OF  LEADING  EXPERIMENTS. 


of  the  electroscope  are  in  the  field  of  the  lantern,  and  the 
metal  plate  P  of  the  electroscope  is  connected  by  a  fine  wire  to  an 
insulated  metal  plate  P'  which  stands  on  the  lecture  table.  The 
fine  wire  should  be  held  loosely  by  small  weights  as  shown,  and 
the  insulating  supports  S  and  S'  should  be  blocks  of  cast 


S 

S 

Y 

—  1  — 

\ 

K 

1 

\ 

M 

7 

\' 

1 

Fig.  58. 

sulphur  and  their  surfaces  should  be  freshly  scraped.  The 
support  5  should  be  made  of  four  brick-like  slabs  (as  shown  in 
section)  thus  shielding  the  gold  leaves  from  currents  of  air  that 
might  enter  the  case  around  the  rod  R.  MM  are  metal  strips 
connected  to  earth.  Of  course  the  case  has  glass  front  and  back. 
In  using  a  gold  leaf  electroscope  it  is  usually  most  convenient 
to  charge  the  electroscope  by  influence,  using  a  glass  rod  which  is 
positively  charged  by  rubbing  it  with  silk,  or  a  hard-rubber  rod 
which  is  negatively  charged  by  rubbing  it  with  fur. 

109.  Charging  by  influence. — Bring  a  positively  charged  glass 
rod  near  plate  P',  Fig.  58,  touch  P'  with  the  finger  and  with- 
draw the  charged  glass  rod,  and  the  entire  system  PP'  will  be 
left  negatively  charged. 

no.  Use  of  the  gold-leaf  electroscope. — (a)  When  the  system 
PP',  Fig.  58,  is  neutral  a  positive  or  a  negative  charge  brought 
near  to  P'  will  cause  the  gold  leaves  to  spread  apart. 

(b)  When  the  system    PP'    is  charged,  a  like  charge  brought 
near  to   P'   increases  the  spread  of  the  leaves. 

(c)  When  the  system    PP'    is  charged,  an  unlike  charge,  if 
large  enough,  brought  nearer  and  nearer  to  P'  lessens  the  spread 
of  the  leaves  to  zero  and  then  causes  the  leaves  to  spread  apart 
again  as  it  comes  still  nearer  to  P'. 


ELECTRICITY  AND   MAGNETISM.  135 

Note. — These  effects  should  be  explained,  let  us  say,  in  terms 
of  the  two  fluid  theory  so  as  to  enable  the  student  to  hold  them 
in  mind  for  a  while  at  least.  The  behavior  of  the  gold-leaf 
electroscope  is  complicated  by  changes  of  capacity  as  may  be 
understood  from  experiment  114,  and  therefore  the  statements 
under  (b)  and  (c)  should  in  all  strictness  be  qualified. 

in.  Sharing  of  charge  by  two  insulated  metal  balls  brought 
into  contact. — Take  two  similar  metal  balls  4  or  5  inches  in 
diameter  with  insulating  handles.  Charge  one  ball  positively, 
let  us  say,  note  its  effect  on  gold  leaves  when  it  is  brought  near 
to  Pf  in  Fig.  58  (electroscope  being  already  slightly  charged), 
touch  it  to  the  other  ball,  and  then  show  that  the  two  balls  have 
nearly  equal  positive  charges,  and  that  the  charge  on  each  ball 
is  considerably  less  than  what  was  initially  on  the  first  ball. 

Note. — The  greatest  amount  of  charge  that  can  be  held  on  a 
5-inch  metal  ball  in  air  would  produce  about  a  thirty-thousandth 
of  an  ampere  for  a  thousandth  of  a  second  and  therefore  one 
cannot  easily  study  the  transfer  of  charge  from  one  ball  to  the 
other  by  measuring  the  electric  current  which  is  associated  with 
(which  indeed  constitutes)  the  transfer  of  charge. 

112.  Giving  up  of  entire  charge  by  internal  contact. — Charge 
a  metal  ball  and  touch  it  to  the  inside  of  a  metal  can  which  is 
itself  supported  by  an  insulating  handle,  and  show  that  the  ball 
is  left  almost  entirely  without  charge  whereas  the  charge  on 
the  can  increases  more  and  more  with  each  internal  contact  of 
the  freshly  charged  ball. 

113.  The  electric  doubler. — The  experiment  which  is  described 
in  Art.  221,  page  315,  General  Physics,  is  very  interesting  and 
instructive.     Strands  of  loose  cotton  threads  serve  very  well  in 
place  of  the  slabs  of  pith.     The  threads  should  be  drawn  re- 
peatedly between  the  fingers  to  make  them  slightly  conducting. 

It  is  instructive  to  place  the  metal  balls  in  the  position  shown 
in  Fig.  254,  page  315,  General  Physics,  and  show  that  they  have 
approximately  equal  but  opposite  charges  when  they  are  sepa- 


136 


CALENDAR  OF  LEADING  EXPERIMENTS. 


rated    and    withdrawn.     This    experiment    shows    charging    by 
influence  in  its  most  complete  and  simplest  form. 

114.  Change  of  capacity  of  plate  P'  in  Fig.  58  and  its  effect 
on  the  gold  leaves.  Demonstration  of  inductivity. — 

Two  elastic  bags  A  and  A'  Two  insulated  metal  bodies 
are  connected  by  a  pipe,  and  air  P  and  P'  are  connected  by  a 


P   fl 


wire 


ft     P' 


Fig.  59. 

is  pumped  into  the  bags  so  as 
to  inflate  them. 

If  the  walls  of  bag  A'  could 
be  weakened  by  making  them 
thinner  some  of  the  air  would 
flow  out  of  A  into  A',  thus 
reducing  the  amount  of  air  in 
A. 


If  the  material  of  bag  A' 
could  be  made  of  more  yielding 
stuff  (without  changing  its 
thickness)  some  of  the  air  would 
flow  from  A  into  A'. 


Fig.  60. 

wire   and    charged    with    elec- 
tricity. 

If  the  dielectric  which  sur- 
rounds P'  is  made  more  yield- 
ing by  making  it  thinner  some 
of  the  charge  will  flow  out  of  P 
into  P',  thus  reducing  the 
amount  of  charge  on  P  and 
decreasing  the  spread  of  the 
gold  leaves.  By  bringing  a 
grounded  metal  body  near  to 
P'  as  shown  the  specified  effect 
is  produced. 

If  a  slab  of  paraffine  is  placed 
between  P'  and  B  the  air 
dielectric  is  replaced  to  some 
extent  by  a  more  yielding 
dielectric  (a  dielectric  with 
greater  inductivity)  and  some 
of  the  charge  will  flow  out  of 
P  into  P',  thus  reducing  the 
spread  of  the  gold  leaves. 


ELECTRICITY  AND   MAGNETISM.  137 

115.  The  corona  discharge  may  be  shown  in  a  dark  room  by 
stretching  two  very  fine  wires  two  to  six  inches  apart  and  con- 
necting them  to  an  influence  electric  machine  or  to  the  secondary 
coil   of  a   step-up   transformer  giving  a   secondary  voltage  of 
twenty  or  thirty  thousand  volts. 

The  familiar  old-time  name  of  the  corona  discharge  is  brush 
discharge,  and  it  is  seen  on  the  projecting  points  on  an  influ- 
ence machine  which  is  operated  in  a  dark  room. 

116.  Electrical   smoke    deposition. — This   experiment   is   de- 
scribed and  explained  in  Art.  225,  page  319,  General  Physics. 

117.  The  ozonizer  is  described  and  its  action  is  explained  in 
Art.  226,  page  320,  General  Physics. 

Discharge  of  electricity  through  gases  and  radio-activity. — 

One  of  the  best  teachers  of  elementary  physics  in  the  United 
States,  who  is  well  known  for  the  splendid  research  he  has  done 
in  recent  years,  once  remarked  to  the  authors  that  he  never 
could  find  the  time  to  devote  to  the  discharge  of  electricity 
through  gases  and  to  the  atomic  theory  of  electricity  in  his 
rather  severe  course  in  elementary  physics  (the  authors  had 
admitted  their  own  difficulties  in  this  respect) ;  but,  he  added, 
"I  often  have  a  spare  lecture  hour  or  two  near  the  end  of  the 
term  which  enables  me  to  show  and  discuss  a  few  experiments, 
and,"  he  added,  in  a  highly  significant  semi-humorous  vein, 
'  I  generally  prepare  myself  for  these  lectures  by  looking  up  the 
subject  in 's  High  School  Physics." 

Following  is  a  list  of  experiments  which  we  have  found  satis- 
factory : 

(a)  Show  the  decreasing  dielectric  strength  of  a  gas  (air)  with 
decreasing  pressure  by  connecting  a  vacuum  tube  in  parallel 
with  an  adjustable  spark  gap,  and  exhausting  the  vacuum  tube 
slowly. 

(&)  Show  the  characteristic  Geissler  tube  discharge. 

(c)  Show  the  characteristic  Crookes  tube  discharge. 

(d)  Exhibit  cathode  ray  shadow  and  show  magnetic  deflection 


138  CALENDAR  OF   LEADING   EXPERIMENTS. 

of  cathode  rays.  Note:  Cathode  rays  and  canal  rays  are  both 
shown  by  the  vacuum  tube  devised  by  Professor  C.  T.  Knipp, 
of  the  University  of  Illinois.  See  Science,  Vol.  XLII,  page  942, 
December  31,  1915. 

(e)  Show  luminescence  produced  by  cathode  rays.  Common 
forms  of  Crookes  tubes  for  this  purpose  are  (i)  A  Crookes  tube 
in  which  several  varieties  of  minerals  are  mounted  and  (2)  A 
Crookes  tube  in  which  the  back  of  a  metal  butterfly  is  painted 
with  several  varieties  of  finely  crushed  minerals. 

(/)  Give  a  demonstration  of  X-rays  using  a  fluoroscope. 
It  is  advisable  for  the  operator  to  shield  himself  from  X-rays  by 
using  screens  of  sheet  lead. 

(g)  Show  the  ionizing  action  of  X-rays  by  holding  a  Crookes 
tube  so  that  the  X-rays  may  pass  near  to  the  plate  P',  Fig.  58. 

(h)  Show  the  ionizing  action  of  the  a-rays,  /3-rays  and  7-rays 
from  radium  by  holding  near  P',  Fig.  58,  a  metal  plate  upon 
which  a  minute  quantity  of  radium  has  been  deposited.  This 
plate  is  to  be  kept  in  a  small  box  when  it  is  not  in  use. 

(i)  Exhibit  the  spinthariscope. 

(j)  Exhibit  the  paths  in  a  fog  chamber  of  individual  a-particles 
from  radium.  A  fog  chamber  for  this  purpose  is  for  sale  by 
The  Cambridge  Scientific  Instrument  Company,  Cambridge, 
England. 


PART    IV. 
LIGHT. 


139 


A  NEW  ENGINE,  OR  A  HELP  TO  THE  MIND  CORRESPONDING  TO 
TOOLS  FOR  THE  HAND. 

Bacon  listed  long  ago,  in  his  quaint  way,  the  things  which  seemed  to  him  most 
needful  for  the  advancement  of  learning,  and  among  other  things  he  mentioned 
A  New  Engine,  or  a  Help  for  the  Mind  corresponding  to  tools  for  the  Hand;  and 
the  most  remarkable  aspect  of  present-day  physical  science  is  the  aspect  in  which  it 
constitutes  a  realization  of  Bacon's  New  Engine.  We  now  force  upon  extremely 
meager  data  (obtained  directly  through  our  senses)  interpretations  which  would 
seem  to  be  entirely  incommensurate*  with  the  data  themselves,  and  we  exercise 
over  physical  things  a  kind  of  rational  control  which  greatly  transcends  the  native 
cunning  of  the  hand.  The  possibility  of  this  forced  interpretation  and  of  this 
rational  control  depends  upon  the  use  of  two  complexes. 

(a)  A  logical  structure,  that  is  to  say,  a  body  of  mathematical  and  conceptual 
theory  which  is  brought  to  bear  upon  the  immediate  materials  of  sense,  and 

(&)  A  mechanical  structure,  that  is  to  say,  either  a  carefully  planned  arrangement 
of  devices  or  a  carefully  planned  order  of  operations  such  as  the  successive  operations 
of  solution,  reaction,  filtration,  and  weighing  in  chemistry. 

These  two  complexes  do  indeed  constitute  a  New  Engine  for  the  Mind,  and  the 
study  of  elementary  physical  science  is  intended  to  lead  to  the  realization  of  this 
Engine  (i)  By  the  building  up  of  the  logical  structure  in  the  mind  of  the  student, 
(2)  By  training  in  the  use  of  devices,  as  in  the  making  of  measurements,  and  in  the 
performance  of  ordered  operations,  and  (3)  By  exercises  in  the  application  of  (i) 
and  (2)  to  the  actual  phenomena  of  physics  and  chemistry  at  every  step  and  all  of 
the  time  with  every  possible  variation. 

That  surely  is  an  exacting  program;  but  the  only  alternative  is  to  place  the 
student  under  the  instruction  of  Jules  Verne  where  he  need  not  trouble  himself 
about  foundations,  but  may  follow  his  teacher  pleasantly  on  a  care-free  trip  to  the 
moon  or  with  easy  improvidence  embark  on  a  voyage  of  twenty  thousand  leagues 
under  the  sea. 

From  The  Study  of  Science,  an  introduction  to  our  Mechanics  and  Heat,  The 
Macmillan  Co.,  1910;  reprinted  in  Bill's  School  and  Mine  by  W.  S.  Franklin, 
published  by  Franklin,  MacNutt  and  Charles,  South  Bethlehem,  Pa. 

*  An  astronomer,  for  example,  looks  at  a  spec  of  light  as  it  crosses  the  field  of  his 
telescope,  and  he  listens  to  the  ticks  of  a  clock  to  note  the  time  of  day  when  the  spec 
crosses  the  center  line  of  the  field.  He  then  examines  a  set  of  fine  lines  on  a  divided 
circle  to  find  the  angular  altitude  of  the  spec  above  the  horizon.  This  he  does  three 
times  in  succession.  Then  he  proceeds  to  calculate  when  the  spec  (a  comet)  will 
be  nearest  the  sun,  how  far  it  will  then  be  from  the  sun,  how  fast  it  will  be  moving, 
and  when  it  will  return,  maybe  a  hundred  years  later! 


I40 


LIGHT. 

118.  Unit  sensory  areas  on  the  skin. — A  very  interesting 
experiment  is  to  blind-fold  an  assistant  and  apply  the  points 
of  a  compass  to  finger  tips,  to  back  of  hand,  to  arm  and  to  the 
back  of  the  neck,  and  bring  the  points  of  the  compass  closer 
and  closer  together  until  in  each  case  the  two  points  can  no 
longer  be  distinguished  as  two. 

The  approximate  diameter  of  the  unit  areas  in  the  central 
part  of  the  retina  can  be  determined  by  placing  two  minute 
globules  of  mercury  very  close  together  (distance  apart  being 
measured)  in  bright  sunlight,  and  finding  greatest  distance  from 
the  eye  at  which  the  two  points  of  light  are  distinguishable 
as  two  points. 

119.  The  astigmatic  pencil  of  rays.— A  clear  understanding  of 
the  difference  between  the  homocentric  pencil  of  rays  and  the 
astigmatic  pencil  of  rays  is  necessary  if  one 

wishes  to  understand  the  important  subject 
of  lens  imperfections. 

A  large  wooden  torus  or  ring  is  very  help- 
ful in  the  discussion  of  Fig.  276,  page  357, 
General  Physics. .  Also  the  wire  model 
which  is  shown  in  Fig.  61  is  very  helpful. 
The  wire  AA  forms  an  arc  of  a  circle  of  Fig  61 

which  the  center  is  at  C.  By  rotating  the  Heavy  lines  are  stiff 
model  through  a  small  angle  about  DD  as  wires,  light  lines  are  fine 

strings. 

an  axis,  the  arc   AA  describes  a  portion  of 

a  ring  surface;  thinking  of  this  surface  as  a  wave  front,  the 
successive  positions  of  the  strings  represent  rays,  and  these  rays 
intersect  along  the  line  C  (perpendicular  to  the  plane  of  the 
paper)  and  along  the  line  DD. 

141 


142  CALENDAR  OF  LEADING  EXPERIMENTS. 

Note. — A  narrow  astigmatic  pencil  of  rays  may  be  obtained 
by  passing  a  beam  of  light  from  an  electric  arc  obliquely  through 
the  central  portions  of  a  long-focus  converging  lens,  as  indicated 
in  Fig.  277,  page  358,  General  Physics,  and  the  linear  foci  at  0 
and  at  DD  may  be  thrown  upon  a  small  piece  of  cardboard 
or  on  a  translucent  screen.  The  central  portion,  only,  of  the 
lens  must  be  used,  that  is,  the  lens  must  be  covered  with  an 
opaque  disk  with  a  hole  at  its  center,  otherwise  the  astigmatic 
effect  is  hidden  by  the  complicated  effects  of  the  coma.  See 
page  400,  General  Physics.  A  much  broader  astigmatic  pencil, 
not  perceptibly  confused  by  coma,  can  be  obtained  by  reflecting 
the  light  from  an  electric  arc  (or  the  light  from  the  sun)  obliquely 
from  a  concave  mirror. 

1 20.  Total  reflection. — The  best  demonstration  of  total  reflec- 
tion is  to  pass  around  the  class  several  pieces  of  plate  glass 
asking  the  students  to  note  the  silvery  appearance  of  the  squarely 
cut  edge  as  seen  by  looking  obliquely  through  the  glass  (showing 
how  to  look  while  making  the  statement) ;    also  several  small 
45°-45°-9O°  glass  prisms  may  be  used. 

A  brilliant  beam  of  light  may  be  passed  through  a  tank  of  clear 
water  and  focused  in  a  small  aperture  through  which  the  water 
issues  as  a  jet.  The  beam  of  light  is  kept  in  the  smooth  jet  by 
total  reflection  until  the  surface  of  the  jet  becomes  deeply  rippled 
or  breaks  into  drops.  This  experiment,  which,  it  must  be 
admitted,  is  of  little  value,  may  be  modified  by  using  a  bent 
glass  rod  instead  of  the  water  jet. 

121.  Refraction  of  a  spherical  wave  at  a  plane  surface. — A 
very   interesting  and   instructive   experiment,   which   must   be 
carried  out  by  each  student  for  himself,  is  as  follows:  The  point 
0,    Fig.  62,  is  a  small  bright  object  (a  bit  of  chalk  or  a  globule 
of  mercury)  on  the  blackened  bottom  of  a  basin  of  clear  still 
water.     Looking  vertically  downwards  at    O    it  appears  to  be 
at    F    (the  center  of  curvature  of  the  portion  of  the  refracted 
wave    WW    at  or  near    V).     Looking  along  the  line    aa    DD, 


LIGHT.  143 

the  point  0  appears  to  be  at  DD  if  the  line  joining  the  two 
eyes  is  horizontal,  but  the  point  0  appears  to  be  at  C  if  the 
line  joining  the  two  eyes  is  vertical.  The  pencil  of  rays  at  aa 
is  astigmatic. 


AIR 


AIR 


GLASS 


Fig.  62. 

122.  The  actual  lens  and  the  ideal  lens. — Pass  a  beam  of  light 
from  an  electric  arc  through  a  large,  short-focus  lens,  such  a 
lens,  for  example,  as  is  used  in  the  condenser  of  a  projection 
lantern,  and  render  the  convergent  beam  of  light  beyond  the 
lens  visible  by  a  cloud  of  chalk  dust  (using  black  board  erasers) 
or  by  a  cloud  of  ammonium  chloride  smoke.     The  appearance  of 
the  beam  is  shown  in  Fig.  342,  page  397,  General  Physics,  and 
described  in  Art.  287,  page  397.     This  beam  becomes  very  much 
more  complicated  if  it  passes  obliquely  through  the  lens. 

Point  out  the  fact  that  the  whole  theory  of  the  simple  lens  as 
developed  in  Chapter  XX,  General  Physics,  applies  to  the  ideal 
simple  lens  as  explained  in  Art.  267,  pages  373-374,  General 
Physics. 

123.  Model  of  a  compound  microscope. — Mount  as  an  object 
a  small  flat  piece  of  fine  wire  gauze  A ,   as  object  glass  a  one-inch 


144  CALENDAR  OF  LEADING  EXPERIMENTS. 

focal  length  lens  0,  and  as  an  eye  piece  a  longer  focal-length 
lens  E  on  a  supporting  bar  as  indicated  in  Fig.  63.  Have  this 
model  in  hand  while  explaining  Art.  277,  page  386,  General 
Physics,  and  then  pass  it  around  the  class. 


axis 


Fig.  63.  Fig.  64. 

124.  Model  of  a  simple  telescope. — Mount  as  object  glass  a 
long  focal  length  lens    O   and  as  eye  piece  a  shorter  focal  length 
lens   £   on  a  supporting  bar  as  indicated  in  Fig.  64.     Have  this 
model  in  hand  and  use  it  to  look  at  a  distant  object  through  an 
open  window  while  explaining  Art.  278,  pages  387-388,  General 
Physics;  and  give  to  each  student  an  opportunity  to  use  it. 

125.  Experiment  on  vision. — We  base  our  sense  of  distance  of 
an  object  (a)  On  a  knowledge  of  the  size  of  the  object  as  com- 
pared with  its  apparent  size,  thus  the  finger  at  arm's  length 
looks  larger  than,  and  can  be  made  to  cover,  a  man  at  a  hundred 
yards,     (b)  By  the  muscular  sense  associated  with  the  turning 
of  both  eyes  directly  towards  the  object,  and  (c)  By  the  muscular 
sense  associated  with  accommodation    (see  page  382,   General 
Physics).     If  we   look  at  an   unfamiliar  object  with   one   eye 
through  a  pin  hole  held  near  the  eye,  bases  (a)  and  (b)  evidently 
fail,  and  base  (c)  also  fails  because  accommodation  is  almost 
unnecessary  when  we  look  through  a  pin  hole. 

Stretch  a  cord  across  the  room,  using  a  kind  of  cord  which 
comes  in  a  wide  range  of  sizes  with  same  kind  of  twist.  Looking 
at  the  cord  with  one  eye  through  a  pin  hole  walk  towards  it, 
then  stop,  and  bring  the  fingers  straight  up  from  below  as  if  to 
pinch  the  cord!  The  result  is  very  amusing.  The  experiment 
works  quite  satisfactorily  even  when  one  has  handled  the  cord 
and  is  familiar  with  its  size.  Sidewise  or  up-and-down  move- 


LIGHT.  145 

ments  of  the  head  tend  to  supply  the  missing  sense  of  distance 
which  is  dependent  on  two-eye  vision,  and  therefore  the  cord 
should  be  on  a  level  with  the  experimenter's  eyes  and  as  the 
experimenter  moves  towards  the  cord  he  should  not  raise  or 
lower  his  head  nor  move  it  sidewise. 

The  absence  of  any  need  for  accommodation  when  one  looks 
through  a  fine  pin  hole  can  be  shown  by  looking  through  such 
a  hole  (held  very  close  to  the  eye)  at  a  printed  page  or  preferably 
at  a  piece  of  wire  gauze  and  bringing  the  gauze  closer  and  closer 
to  the  eye. 

126.  Experiment  on  vision.  The  coin  box. — Nothing,  perhaps, 
is  more  familiar  than  a  silver  coin,  and  in  a  certain  United  States 
coinage  one  face  is  exactly  similar  in  the  dollar,  the  half-dollar, 
the  quarter-dollar  and  the  dime,  and  a  very  amusing  device  is 
a  box  with  a  fine  peep  hole  at  one  end  with  these  similar  coins 
arranged  at  distances  from  the  peep  hole  proportional  to  their 
respective  diameters.  The  coins  should  be  supported  on  pedes- 
tals the  widths  of  which  are  in  proportion  to  the  diameters  of 
the  coins,  and  the  pedestals  should  be  high  enough  (box  deep 
enough)  so  that  the  bottom  of  the  box  can  be  made  invisible  by 
a  small  screen  at  a  short  distance  from  the  peep  hole. 

We,  the  authors,  find  that  the  coin  box  is  almost  sure  to  get 
out  of  order  from  one  year  to  the  next,  because  the  coins  get 
loose  and  are  lost ;  but  such  things  have  never  greatly  disturbed 
us.  We  manage  to  give  our  best  efforts  to  our  students  in  spite 
of  the  ever  insistent  call  of  pessimism,  and,  as  Professor  Wm. 
Lyon  Phelps  says,  teaching  is  great  fun.  But  we  know  teachers, 
college  teachers,  who  for  such  reasons,  and  often,  alas,  for 
reasons  much  less  respectable  and  adequate,  never  take  any 
pains  at  all!  Professor  Phelps  has  the  advantage  of  us  in  that 
he  teaches  English.  What  unlimited  opportunity  for  humor  an 
English  teacher  has,  and  how  shameful  it  is  that  English  teaching 
should  ever  have  the  fatal  quality  of  dullness!  shameful  even  in 
the  teacher  of  "crude  Pennsylvania  Dutch  boys,"  who,  according 
to  one  English  teacher's  comfortable  belief,  are  unworthy  of  any 
ii 


146  CALENDAR   OF   LEADING   EXPERIMENTS. 

effort  on  his  part! — We  can  say  no  more,  lest  "the  gals  hold  their 
fans  before  their  faces"  as  they  did  at  Miss  Meadow's  party 
when  Ole  Brer  Tarrypin  led  them  to  think  that  Brer  Fox  had 
cussed  when  he  met  him  on  the  big  road.  Brer  Fox  hadn't 
cussed,  he  had  merely  said,  Hello  Stinkin  Jim.  Think  of  calling 
a  nice  man,  like  Brer  Tarrypin,  Stinkin  Jim!  His  only  fault  is 
that  he  is  constitutionally  slow,  but  his  patience  is  illustrated 
by  what  he  tells  Brer  Fox  about  his  being  caught  in  a  fire- 
swept  field.  Well,  but  what  did  you  do  Brer  Tarrypin?  Sot 
and  tuck  'er,  Brer  Fox,  sot  and  tuck  'er.  • 

One  of  the  most  remarkable  things  in  this  world  is  the  settled 
character  of  a  teacher's  position.  Where  a  teacher  begins  he 
usually  stays  for  life;  and  a  teacher  should  make  of  his  im- 
mobility an  opportunity  for  virtue,  like  Ole  Brer  Tarrypin,  even 
to  the  extent  of  teaching  English  to  "crude  Pennsylvania  Dutch 
boys " ;  for  all  boys  are  crude;  yes,  even  the  far  away  groups  that 
— that — never  bother  us,  let  us  say. 

127.  Experiment  on  vision.  Why  is  an  object  seen  erect  when 
its  image  on  the  retina  is  inverted? — In  answer  to  this  question 
the  equally  sensible  question  is  sometimes  asked:  When  one 
hears  a  baby  cry  with  two  ears,  why  does  one  not  take  it  for 
twins?  Our  sensations  are  symbols  which  we  interpret  in  ac- 
cordance with  experience. 

Make  a  very  small  kink  or  curl  at  the  end  of  a  bit  of  very  fine 
wire,  hold  the  curl  near  the  eye,  beyond  the  curl  hold  a  fine  pin 
hole  and  look  towards  a  broad  bright  surface.  An  erect  shadow 
(an  erect  image)  of  the  curl  is  thrown  on  the  retina,  and  the  curl 
is  seen  inverted.  If  we  always  looked  at  curls  in  this  way 
we  would  develop  the  habit  of  seeing  them  right  side  up. 

The  compound  microscope  and  the  astronomical  telescope 
always  invert  an  object  which  is  being  examined,  and  this 
inversion  is  confusing  to  a  novice,  but  to  the  experienced  astron- 
omer or  microscopist  it  is  not  in  the  least  confusing  or  even 
troublesome. 


LIGHT.  147 

128.  Experiment  on  vision.     Curious  effect  of  two-eye  vision. 

— Stretch  a  thread  or  string  on 

— »          thread 
a  frame  as  indicated  in  Fig.  65, 

and  look  at  the  string  with  the 
two  eyes  as  indicated  by  the  Fig  65 

arrow.     Bring  the  eyes  to  focus 

on  any  point  p  of  the  string  and  one  will  see  two  strings  inter- 
secting at  the  point  p. 

129.  Shadows  of  blood  veins  on  the  retina. — Any  persistent 
phase  of  retinal  excitation  which   is   independent  of  external 
objects  and  conditions  is  by  long  habit  ignored  and  therefore 
imperceptible.     For  example,  a  complex  system  of  arteries  and 
veins  lies  in  front  of  the  retina  and  casts  shadows  on  the  retina, 
but  ordinarily  these  shadows  are  not  perceptible.     However,  if 
the  image  of.  a  kerosene  lamp  flame  is  thrown  by  a  moderately 
short  focus  lens  on  the  eye  ball  (eye  being  turned  away  from 
direction  of  lamp  so  that  image  may  fall  on  the  eye  ball  at  some 
distance  to  one  side  of  the  cornea)  in  a  darkened  room,  the  small 
quantity  of  light  which  penetrates  the  thick  walls  of  the  eye 
casts  shadows  of  arteries  and  veins' in  an  unusual  manner  and  the 
entire  net-work  of  veins  and  arteries  becomes  visible. 

130.  Persistence  of  vision  and  the  stroboscope. — The  light 
sensation  produced  by  a  flash  of  light  lasts  from  one  half  of  a 
second  to  one  second  or  more,  according  to  the  intensity  of  the 
light,  and  an  intermittent  light  is  sensibly  continuous  when  the 
flashes  recur  at  a  frequency  of  15  or  20  flashes  per  second  or  more. 

Move  the  hand  rapidly  to  and  fro,  with  outspread  fingers, 
in  the  light  of  an  alternating-current  electric  arc,  and  many 
duplicate  images  of  the  hand  are  seen  because  of  the  inter- 
mittent character  of  the  light  from  such  an  arc. 

Mount  a  slotted  metal  disk  on  the  shaft  of  a  small  electric 
motor  and  arrange  an  electric  arc  (direct-current  arc)  near  the 
edge  of  the  disk  so  that  flashes  of  light  may  pass  through  the 
slot  or  slots  as  the  disk  rotates.  A  cardboard  disk  with  figures 


148 


CALENDAR  OF   LEADING  EXPERIMENTS. 


on  it  marked  in  broad  black  lines  like  Fig.  66  is  mounted  on  a 
second  motor,  the  cardboard  disk  is  illuminated  by  the  flashes 
of  light,  and  the  speed  of  one  or  both 
motors  is  adjusted  until  the  ratio  of  the 
motor  speeds  is  an  integer.  With  the 

OS~*\  \  particular  marking  shown  in  Fig.  66 
\^t  j  the  widest  range  of  effects  can  be  ob- 
tained if  the  slotted  metal  disk  has  four 
equidistant  slots,  and  with  a  10  or  12- 
inch  disk  the  slots  should  be  about  J  an 
inch  wide. 

A  most  amusing  experiment  is  to  use  a  cardboard  disk  marked 
as  indicated  in  Fig.  67. 


Fig.  66. 


Fig.  67. 

131.  Reversal  of  a  sense  of  fatigue. — In  many  cases  it  seems 
that  an  even  blend  of  excitation  of  many  nerve  elements  gives, 
let  us  say,  a  neutral  or  colorless  sensation,  whereas  the  predomi- 
nance of  excitation  of  certain  elements  of  the  blend  gives  a  sharp 
and  distinctive  character  to  the  corresponding  sensation.  This 
is  exemplified  by  the  sensations  of  color  according  to  the  Young- 
Helmholtz  theory  (see  General  Physics,  pages  479-481),  and  it 


LIGHT.  149 

is  also  shown  by  the  reversal  by  fatigue  of  the  sense  of  motion. 
When  one  looks  at  the  floor  after  looking  for  a  long  time  out  of 
the  window  of  a  moving  railway  car,  one  sees  the  floor  in  motion 
in  a  reversed  direction.  Mark  a  broad  black  spiral  on  a  card- 
board disk,  rotate  the  disk  steadily  at  a  speed  of  three  or  four 
revolutions  per  second,  look  intently  at  the  rotating  disk  for 
half-a-minute  or  more,  and  then  turn  and  look  at  the  face  of  a 
nearby  companion.  Your  companion's  head  will  be  seen  to  swell 
or  shrink  according  to  the  direction  of  motion  of  the  disk  and 
spiral. 

This  reversal  of  the  sense  of  direction  by  fatigue  seems  to  show 
that  all  the  nerve  excitation  which  is  necessary  for  a  sense  of 
motion  in  any  direction  is  present  in  a  complex  blend  when  we 
look  at  a  stationary  field;  but  that  steady  gazing  at  a  field 
moving  in  one  direction  produces  fatigue  which  results  in  the 
weakening  of  a  certain  element  of  excitation  in  what  would 
ordinarily  be  a  neutral  or  colorless  blend  when  we  look  again 
at  a  stationary  field.* 

132.  Imperfection  of  focal  spot  of  a  perfect  lens. — By  passing 
the  light  from  a  distant  electric  arc  through  a  very  fine  pinhole 
p,  Fig.  68,  and  looking  through         , 

the  magnifying  glass    M,    the  ^ 

distribution  of  the  light  on  the  »v 

plane    FF   can    be    seen.      If  > 

the  pin  hole  is  very  fine,  the       ^ 

distribution  of  light  gives  an  Fig  6g 

exaggerated   example    of    the 

imperfections  of  the  focal  spot  of  a  perfect  lens  as  explained  in 
Art.  283,  pages  392-394,  General  Physics. 

133.  Axial  spherical  aberration. — See  experiment  122. 

134.  Astigmatism  of  a  lens  of  narrow  aperture. — (a)  See  note 
under  experiment  119. 

*  This  matter  is  discussed  in  a  brief  note  in  Science,  Vol.  IX,  page  70,  Jan.  13 
1899. 


150  CALENDAR  OF  LEADING  EXPERIMENTS. 

(6)  The  astigmatism  of  a  lens  may  be  shown  by  looking 
obliquely  through  a  low  power  magnifying  glass  at  the  cross- 
rulings  on  cross-section  paper,  the  lens  being  held  so  that  its 
central  plane  intersects  the  plane  of  the  paper  in  a  line  parallel 
to  one  set  of  rulings.  In  this  case  the  pupil  of  the  eye  limits  the 
effective  beam  of  light  to  a  very  narrow  pencil,  and  one  or  the 
other  sets  of  cross  rulings  may  be  sharply  focused  by  adjusting 
the  distance  of  the  lens  from  the  paper. 

(c)  Substitute  a  simple  lens  for  the  object  lens  of  a  projection 
lantern,  cover  the  lens  with  an  opaque  disk  with  a  hole  at  its 
center,  turn  the  lens  so  that  the  light  passes  obliquely  through  it, 
and  use  a  piece  of  wire  gauze  as  a  lantern  slide.  By  adjusting 
the  distance  of  the  oblique  simple  lens  from  the  gauze  the  vertical 
or  horizontal  wires  may  be  focused  on  the  lantern  screen.  Of 
course  the  central  plane  of  the  oblique  lens  must  intersect  the 
plane  of  the  gauze  along  a  line  parallel  to  one  set  of  wires. 

135.  Image   distortion. — Using  a  piece  of  wire  gauze  as  a 
lantern  slide,  and  a  simple  lens  as  the  object  lens  of  the  lantern, 
the  two  kinds  of  image  distortion  can  be  shown  as  explained  in 
Art.  291,  pages  402-403,  General  Physics.     It  is  worth  while 
also  to  use  the  regular  orthoscopic  lantern  objective  and  to  pro- 
ject an  undistorted  image  of  the  gauze  on  the  lantern  screen. 

136.  Chromatic  aberration. — The  image  of  the  wire  gauze  as 
projected  by  the  simple  lens  in  experiment  135  shows  traces  of 
color  due  to  the  chromatic  aberration  of  the  lens,  but  the  beam 
of  light  passes  chiefly  through  the  central  part  of  the  lens  as 
indicated  in  Figs.  352  and  353,  page  403,  General  Physics,  so 
that  the  chromatic  aberration  is  small  as  explained  in  connection 
with  Fig.  362,  page  411,  General  Physics. 

The  regular  orthoscopic  lantern  objective  is  corrected  for 
achromatic  aberration  and  it  projects  a  nearly  color-free  image 
of  the  wire  gauze  on  the  screen. 

The  lenses  of  the  eye  are  by  no  means  achromatic,  but  we 
have  come  by  habit  to  neglect,  and  therefore  not  to  see,  the  color 


LIGHT. 


fringes  even  of  brilliant  objects.  Any  unusual  condition,  how- 
ever, makes  this  habit  ineffective.  Thus  by  looking  at  a  distant 
brilliant  lamp  a  very  decided  show  of  color  is  seen  as  the  finger 
is  held  near  the  eye  and  moved  slowly  sidewise  until  it  nearly 
covers  the  pupil. 

An  experimental  study  of  chromatic  aberration  suitable  for 
the  laboratory  is  described  in  Franklin,  Crawford  and  MacNutt's 
Practical  Physics,  Vol.  Ill,  The  Macmillan  Co. 

137.  The  telephoto  lens. — Figure  69  represents  the  object  lens 
O  and  the  eye  piece  £  of  a  simple  telescope,  the  eyepiece  being 
drawn  out  so  that  its  distance  from  the  image  i  is  greater  than 


V 


Fig.  69. 


its  focal  length  EF.  Under  these  conditions  an  enlarged  real 
image  of  i  is  formed  at  I.  Figure  70  represents  an  opera-glass 
type  of  telescope  with  the  eyepiece  E  pushed  in  so  that  its 
distance  from  the  image  i  is  greater  than  its  focal  length  EF. 
Under  these  conditions  the  eyepiece  forms  an  enlarged  real  image 
of  i  at  /.  Either  arrangement,  Fig.  69  or  Fig.  70,  can  be  used 
as  a  telephoto  lens  but  the  arrangement  in  Fig.  70,  being  shorter, 
is  more  convenient.  The  individual  lenses  0  and  E  are  always 
compound  lenses.  Thus  the  left-hand  group  of  lenses  in  Fig. 
373,  page  417,  General  Physics,  takes  the  place  of  0  in  Fig.  70, 
and  the  right-hand  group  of  lenses  in  Fig.  373  takes  the  place 
of  E  in  Fig.  70. 

An  experiment  of  great  interest  is  to  use  a  simple  telescope 
as  indicated  in  Fig.  69  for  projecting  an  image  of  the  sun  on  a 
sheet  of  cardboard.  If  a  large  astronomical  telescope  is  not 


152 


CALENDAR  OF  LEADING  EXPERIMENTS. 


available  a  small   reading   telescope   from   the   laboratory  will 
answer.     The  telescope  should  be  fixed  in  a  hole  through  a 


Fig.  70. 

large  board  (to  darken  the  region  back  of  the  telescope),  and 
the  image  should  be  thrown  on  a  translucent  screen  of  ground 
glass  so  as  to  be  easily  seen  by  a  group  of  persons. 

138.  Spectroscope  demonstrations. — To  project  a  large  and 
sharply  defined  spectrum  upon  a  screen  requires  very  expensive 
apparatus  and  therefore  the  authors  always  arrange  several 
laboratory  spectroscopes  so  that  the  members  of  a  class  may  one 
by  one  see  the  following : 

(a)  The  continuous  spectrum  of  a  gas  flame  or  glow  lamp. 

(b)  The  bright-line  spectrum  of  an  alkali  metal  in  a  Bunsen 

flame. 

(c)  The  reversal  of  the  sodium  lines,  and 

(d)  The  dark-line  spectrum  of  the  sun. 

The  reversal  of  the  sodium  lines  may  be  seen  by  looking  through 
a  Bunsen  flame  at  the  crater  of  a  direct-current  carbon  arc  and 
placing  in  the  Bunsen  flame  a  small  roll 
0  of  asbestos  cloth  which  has  been  previ- 

ously soaked  in  strong  brine  and  dried. 

139.  Projection  of  a  soap  film. — The 

interference  colors  of  a  thin  film  may  be 
p.  71  beautifully  shown  by  projection.  A  thin 

metal  disk  dd,  Fig.  71,  with  a  circular 

hole  two  or  three  inches  in  diameter  is  dipped  into  a  soap  solu- 
tion and  placed  as  shown  in  front  of  the  lantern  condenser  CC 


LIGHT.  153 

and  an  image  of  the  soap  film   S   is  projected  on  the  screen  by 
the  lantern  objective    0. 

140.  Spectroscopic  analysis  of  the  light  reflected  from  a  thin 
film. — Arrange  a  thin  film  of  mica  so  as  to  reflect  the  light  from 
a  gas  flame  or  glow  lamp  into  the  slit  of  a  spectroscope.     It  is 
desirable  in  this  experiment  to  use  a  film  too  thick  to  show  inter- 
ference color  to  the  unaided  eye.     The  spectrum  will  show  a 
large  number  of  bright  and  dark  bands,  and  no  color  is  seen 
directly  by  the  unaided  eye  because  the  intensified  wave  lengths 
(and  the  weakened  wave  lengths)  are  distributed  throughout  the 
spectrum.     When  light  from  a  soap  film  is  reflected  into  the  slit 
of  a  spectroscope  one  or  two  parts  of  the  spectrum  are  intensified 
(and  one  or  two  parts  weakened)  at  a  time. 

141.  The  diffraction  grating. — A  most  striking  demonstration 
is  to  look  through  a  glass  grating  (200  or  300  lines  to  the  inch) 
at  an  electric  arc  between  impregnated  carbons.     This  gives  the 
arrangement  shown  in  Fig.  400,  page  440,  General  Physics,    LL 
being  the  lens  of  the  eye  and   PP   being  the  retina. 

142.  Polarized  waves  on  a  rubber  tube. — Tie  one  end  of  a 
rubber  tube  fast  and  hold  the  other  end  in  the  hand  with  the  tube 
stretched,  and  give  the  discussion  of  Art.  311,  page  442,  General 
Physics.     Pass  the  rubber  tube  through  two  narrow  slits,  produce 
irregular  waves  on  the  tube  and  point  out  the  polarizing  effect 
of  the  slit  near  the  hand.     Turn  the  second  slit  so  as  to  be 
parallel  to  the  first  and  then  turn  it  so  as  to  be  at  right  angles 
to  the  first  and  show  the  effects  as  described. 

143.  Group   of   experiments   on   polarized   light. — Polarizing 
apparatus  for  projection  is  expensive  and  the  authors  have  there- 
fore always  arranged  what  our  students  have  called  a  "seven- 
ring  circus"  for  demonstrations  in  polarized  light.     The  demon- 
strations we  use  in  spectroscopy  also  constitute  a  "seven-ring 
circus."     Each  set  of  apparatus  is  usually  duplicated,  and  the 
entire  teaching  staff  of  the  department  usually  serve  as  "ring- 
masters." 


154  CALENDAR  OF   LEADING   EXPERIMENTS. 

(a)  Show  tourmaline  tongs  and  direct  the  observer  to  look 
through  the  tongs  and  turn  one  of  the  crystals  slowly. 

(b)  Look  through  a  Nicol  prism  at  a  bright  varnished  surface 
and  turn  the  prism  slowly. 

(c)  Place  a  small  rhomb  of  Iceland  spar  on  a  sheet  of  paper  on 
which  is  a  small  black  dot  and  turn  the  rhomb  slowly.     Look  at 
the  rhomb  and  dot  through  a  Nicol  prism  and  turn  the  prism 
slowly. 

(d)  Look  through  two  Nicol  prisms  and  turn  one  of  the  prisms 
slowly. 

(e)  Using  a  simple  polariscope  (parallel  beam)  examine  a  plate 
of  mica  or  other  doubly  refracting  crystal,  turning  polarizer  and 
crystal  plate  slowly,  one  and  then  the  other. 

(/)  Using  a  polariscope  (arranged  for  convergent  beam) 
examine  a  plate  of  Iceland  spar  cut  perpendicularly  to  optic 
axis;  also  examine  a  plate  of  a  biaxial  crystal,  aragonite  for 
example. 

(g)  Using  a  simple  polariscope  (parallel  beam)  examine  a 
cube  of  glass  under  stress. 

Note. — The  inexpensive  Norrenberg  polariscope  can  be  used 
for  e  and  g;  and  for  /  a  pair  of  tourmaline  tongs  held  close  to 
the  eye  is  perhaps  most  satisfactory. 

Note. — Experiment  c  can  be  easily  projected.  Use  a  metal 
lantern  slide  with  a  pin  hole  in  it,  and  place  a  rhomb  of  Iceland 
spar  against  the  slide  covering  the  pin  hole,  and  turn  the  rhomb 
slowly. 

Note. — A  large  cardboard  model  of  a  rhomb  of  Iceland  spar 
with  equal  length  edges  and  with  a  rod  set  in  the  position  of  the 
optic  axis  (the  axis  of  symmetry  )is  useful. 

Note. — A  large  block  of  wood  cut  to  the  form  of  a  Nicol  prism 
and  sawed  in  two  like  a  Nicol  prism  is  useful. 

Note. — The  following  experiments  are  very  interesting  but 
inconvenient  and  unsatisfactory  for  general  purposes. 

(h)  Fill  a  tall  glass  jar  with  a  milky  solution  made  by  dropping 


LIGHT.  155 

a  few  drops  of  extremely  dilute  rosin-alcohol  into  water  and 
stirring  vigorously.  Throw  a  beam  of  light  downwards  into  the 
jar  in  a  dark  room  and  look  sidewise  at  the  jar  through  a  Nicol 
prism,  turning  the  prism  slowly.  Throw  a  beam  of  plane 
polarized  light  downwards  into  the  jar,  rotate  the  plane  of 
polarization,  and  look  at  the  jar  from  one  side.  In  this  second 
experiment  a  test  tube  may  be  used  instead  of  a  jar  so  that  a 
small  sized  Nicol  prism  can  be  used  as  a  polarizer. 

(i)  Fill  a  test  tube  with  a  strong  solution  of  syrup  made 
milky  by  the  addition  of  a  small  quantity  of  extremely  dilute 
rosin-alcohol.  Throw  a  beam  of  plane  polarized  light  downwards 
into  the  tube  in  a  dark  room  and  look  at  the  tube  from  one  side. 
The  tube  will  show  faint  helices  of  colored  bands. 

Note. — The  authors  have  always  arranged  for  students  to 
use  the  saccharimeter  in  the  laboratory.  See  Franklin,  Crawford 
and  MacNutt's  Practical  Physics,  Vol.  Ill,  pages  68-73. 

144.  Group  of  experiments  on  color. — 

(a)  Light  from  a  gas  flame  or  glow  lamp  passes  into  the  slit 
of  a  spectroscope  and   the  observer  places  before  the  slit  in 
succession  a  series  of  colored  glasses  or  a  series  of  colored  solutions 
in  narrow  glass  cells. 

(b)  Light  from  a  gas  flame  or  glow  lamp  is  reflected  into  the 
slit  of  a  spectroscope  from  a  series  of  colored  pieces  of  paper  or 
cloth.     It  is  advisable  to  have  a  piece  of  white  paper  placed 
permanently  in  position  to  reflect  light  into  the  spectroscope 
when  the  colored  paper  or  cloth  is  removed. 

(c)  Illuminate  a  batch  of  brightly  colored  objects  (a  selection 
of  small  skeins  of  brilliantly  colored  worsted  such  as  is  used  for 
color-blind  tests)'  by  the  light  from  a  sodium  flame  in  a  dark  room 
and  arrange  an  electric  glow  lamp  so  that  it  can  be  easily  turned 
off  and  on  so  as  to  give  white  light  illumination. 

(d)  Demonstrate    the    familiar    rotating   sector-colored    disk, 
and  adjust  the  sectors  so  as  to  give  a  neutral  gray  resultant  or 
blend. 


156  CALENDAR  OF   LEADING   EXPERIMENTS. 

(e)  Contrast  effect.  Place  side  by  side  on  an  inclined  board 
a  large  sheet  of  green  tinted  paper  and  a  large  sheet  of  red  tinted 
paper  with  good  illumination.  Tear  off  two  bits  of  grayish 
paper  (from  the  same  piece)  placing  one  of  the  bits  on  each 
sheet  of  tinted  paper,  and  tear  off  a  bit  of  each  sheet  of  tinted 
paper  and  place  the  bit  upon  the  other  sheet.  The  colors  of  the 
small  bits  of  paper  are  so  modified  by  the  effects  of  contrast  that 
it  is  almost  impossible  to  believe  that  the  two  bits  of  grayish 
paper  are  the  same,  and  to  believe  that  each  bit  of  colored  paper 
is  the  same  as  the  nearby  large  sheet.  This  experiment  is  most 
effective  if  the  bits  of  paper  are  torn  off  and  placed  after  calling 
the  students'  attention  to  the  experiment  and  while  they  are 
looking  attentively. 

(/)  The  authors  usually  arrange  for  color-blind  tests  of  the 
entire  class.  We  ordinarily  use  the  Holmgren  test,*  and  usually 
we  can  persuade  a  color-blind  person  to  undergo  the  test  before 
the  entire  class. 

*  Colored  worsteds  for  this  test  may  be  obtained  from  E.  B.  Meyrowitz, 
N.  Y.  City. 


PART  V. 
SOUND. 


157 


THE  PHILOSOPHY  OF  STEAM  SHOVELS  AND  THE  PHILOSOPHY  OF 

LIVING. 

Imagine  a  never-to-be-escaped  human  need  of  a  twenty-foot  arm!  What  age- 
long development  and  what  unthinkable  pains!  It  is  easier  to  build  a  steam  shovel. 
All  of  which  means  that  homo  sapiens  is  now  bent  towards  Social  Inheritance;  but 
social  inheritance  has  own  pains  as  many  know  who  burn  the  midnight  oil. 

How  shocking  to  reduce  the  tender-minded  philosopher's  love  of  perfect  pre- 
cision* to  a  materialistic  preference  for  steam  shovels  as  opposed  to  immeasurable 
pains  of  birth;  and  to  make  mathematical  philosophy  appear  as  a  dire  necessity! 
rather  than  as  a  thing  to  be  chosen  for  its  own  sake.  And  then  to  urgej  with  that 
lover  of  paradox,  Gilbert  Chesterton,  that  the  serious  spiritual  and  philosophic 
objection  to  steam  shovels  is  not  that  men  work  at  them  and  pay  for  them  and 
make  them  very  ugly,  nor  even  that  men  are  killed  by  them,  but  merely  that  men 
do  not  play  at  them !  Think  of  a  group  of  sportsmen  cavorting  over  a  ten-thousand 
acre  field  tossing  and  catching  a  Brobdignagian  ball  in  steam  shovels!  It  is 
conceivable  that  the  one  objection  to  the  steam  shovel  might  have  been  overcome 
if  the  Great  War  had  not  come  upon  us. 

The  great  danger  of  our  time  has  been  the  confusion  of  boundaries  between 
thing-philosophy  and  human  philosophy,  between  the  philosophy  of  material 
conquest  and  power  and  that  intimate  philosophy  of  comfort  which  makes  life 
not  easy  but  worth  while.  When  these  boundaries  are  rectified  there  will  be  one 
philosophy  of  steam  shovels,  recognized  and  used  as  such,  and  another  philosophy 
of  living.  Science  will  then  stand  as  the  essence  of  man's  inescapable  responsibilities 
in  practical  affairs,  and  we  shall  seek  God,  a  finite  God,  in  that  which  is  intimately 
and  even  narrowly  human,  if  narrowness  there  be  in  that  supreme  and  illimitable 
mystery. 

From  Education  After  the  War  by  Franklin  and  MacNutt;  reprinted  in  Bill's 
School  and  Mine. 

*  This  is  a  reference  to  the  point  of  view  of  Professor  Keyser  of  Columbia 
University  as  hazily  set  forth  in  his  recent  articles  On  the  Human  Worth  of  Precise 
Thinking.  See  page  34  of  this  volume  for  further  comment. 

t  See  our  Introduction  to  Mechanics  (The  Study  of  Science),  as  referred  to  on 
page  140  of  this  volume. 

J  See  Preface  to  our  Elements  of  Electricity  and  Magnetism,  The  Macmillan 
Co.,  1908. 


158 


SOUND. 

The  senses  of  touch  and  sight  are  preeminently  space  senses, 
as  everyone  knows,  whereas  the  sense  of  sound  might  almost 
be  called  the  time  sense,  because  the  order  in  sequence  of  the 
elements  of  a  given  sound  sensation  is  all-important.  This  is  true 
of  fleeting,  irregular  sounds  as  in  speech,  and  it  is  equally  true  of 
musical  sounds  as  is  shown  by  the  use  of  rythm  and  sequences 
of  tones  in  music. 

145.  An  extremely  amusing  experiment,  and  one  which  illus- 
trates the  importance  of  order  or  sequence  in  the  elements  of  a 
complex  sound  sensation,  is  to  arrange  a  phonograph  so  that  it 
can   be   driven   forwards   or   backwards   at   will   and   produce, 
forwards    and    backwards,    a    familiar    melody    like    "Yankee 
Doodle"  or  a  familiar  speech  like  "Mary  Had  a  Little  Lamb." 

A  melody  or  a  speech  when  reversed  produces,  of  course, 
precisely  the  same  sense  elements  but  nothing  is  more  utterly 
unlike  than  a  melody  and  its  reverse  or  a  speech  and  its  reverse. 

146.  Musical  sticks. — To  show  that  musical  tones  commonly 
are  component  parts  of  many  noises  thump  at  different  points  on 
a  table  or  chair  and  call  attention  to  the  fact  that  the  sound 
differences  are  mostly  differences  of  pitch  in  the  very  brief  tones 
that  are  produced. 

Drop  a  stick  on  the  floor  and  the  sound  is  mostly  noise,  but 
drop  a  carefully  adjusted  series  of  sticks  and  the  tone  differences 
stand  out  by  contrast  as  a  clearly  recognized  melody  or  musical 
scale. 

147.  The  Galton  whistle. — An  interesting  experiment  for  a 
very  small  group  of  listeners  is  to  determine  the  pitch  limit  of 
audibility  of  musical  tones  by  means  of  the  Galton  whistle. 

A  series  of  steel  bars  giving  tones  in  an  ascending  scale  up  to 

159 


160  CALENDAR  OF   LEADING   EXPERIMENTS. 

and  beyond  the  limit  of  audibility  can  be  used.  These  bars  can 
be  cut  from  cold  rolled  steel  shafting  and  if  d  is  the  diameter 
and  /  the  length  of  a  bar  in  inches  the  number  of  complete 
vibrations  per  second  (transverse  vibrations)  corresponding  to 
its  fundamental  mode  is 

d 

n  =  15010^- 

148.  Simple  modes  of  vibration  of  a  string. — Produce  the 
successive  harmonics  on  the  string  of  a  violin  or  guitar.     Using 
a  sonometer  place  paper  riders  on  the  string  and  show  the 
experiment  which  is  described  at  the  top  of  page  509,  General 
Physics. 

149.  Simple  modes  of  vibration  of  an  air  column. — Produce 
the  successive  notes  on  a  bugle.     Using  a  long  narrow  organ 
pipe  blow  it  with  increasing  air  pressure,  and  a  series  of  tones 
corresponding  to  successive  simple  modes  will  be  clearly  heard. 

A  long  glass  tube  an  inch  or  an  inch  and  a  half  in  diameter  is 
arranged  as  a  whistle  or  organ  pipe,  closed  at  one  end  and  blown 
steadily  by  bellows  (the  moist  breath  will  not  do) ,  and  the  vibrat- 
ing segments  are  made  visible  by  means  of  lycopodium  powder. 
Increasing  the  air  pressure  will  cause  the  air  column  to  break  up 
into  shorter  and  shorter  vibrating  segments. 

150.  Simple  modes  of  vibration  of  a  plate.     Chladni's  figures. 
— The  simple  modes  of  vibration  of  a  plate  may  be  shown  as 
explained  on  pages  517-518,  General  Physics. 

151.  Resonance. — (a)  Free  the  strings  of  a  piano  by  pressing 
the  damper  pedal  and  sing  a  series  of  clear  notes  loudly  with  the 
mouth  held  near  to  the  sounding  board.     Each  note  will  set  in 
vibration  that  particular  piano  string  which  vibrates  in  unison 
with  it,  and  each  note  will  be  heard  (as  produced  by  the  vibrating, 
string)  when  the  sung  note  ceases.     A  piano  is  an  important 
part  of  the  equipment  of  a  physics  lecture  room. 

(b)  Hold  a  vibrating  tuning  fork  over  the  mouth  of  a  tall 


SOUND.  161 

slender  glass  jar,  and  pour  water  slowly  into  the  jar.  The 
air  in  the  jar  will  be  set  vibrating  by  the  fork  when  the  frequency 
of  vibration  of  the  air  column  coincides  or  nearly  coincides 
with  the  frequency  of  the  vibrating  fork  and  the  sound  produced 
will  be  greatly  intensified.  This  experiment  is  most  satisfactory 
when  the  water  flows  into  (or  out  of)  the  tall  jar  quietly  through 
a  rubber  tube  connection  at  the  bottom  of  the  jar. 

(c)  An  ordinary  telephone  connected  to  alternating-current 
supply  mains  (through  a  high  resistance,  of  course)  produces  a 
musical  tone  which  is  very  rich  in  over- tones,  and  if  such  a 
singing  telephone  is  held  near  the  mouth  of  the  tall  jar  above 
described  the  successive  over-tones  come  out  with  wonderful 
distinctness  as  the  jar  is  slowly  filled  with  water.  The  over- 
tones may  be  brought  out  also  by  holding  the  singing  telephone 
near  one's  open  mouth  and  changing  the  mouth  cavity  as  if  to 
speak  the  following  vowels  in  succession  u,  5,  a,  a,  e1  and  e  (see 
General  Physics,  page  522).  Ordinary  6o-cycle  alternating  cur- 
rent is  not  very  satisfactory  for  use  in  this  experiment,  but 
the  experiment  is  wonderful  if  one  has  133-cycle  or  i5O-cycle 
current.  One  can  produce  sufficient  alternating  current  by 
an  electrically  driven  tuning  fork,  the  telephone  (very  low  re- 
sistance telephone)  being  connected  between  the  terminals  of 
the  windings  of  the  driven  .magnet,  and,  if  necessary,  a  large 
capacity  condenser  being  connected  across  the  break.  A  tuning 
fork  of  250  or  300  complete  vibrations  per  second  should  be 
used.  An  ordinary  pitch  pipe  can  be  used  instead  of  a  singing 
telephone  but  the  overtones  do  not  come  out  very  sharply. 

152.  Experiments  on  vowel  sounds. — Shape  the  mouth  as  if  to 
produce  the  vowels  u,  5  and  a  in  succession  and  thump  the  cheek, 
and  the  characterizing  tones  (173,  517  and  775  vibrations  per 
second,  respectively,  see  page  522,  General  Physics)  will  be 
distinctly  heard. 

An  extremely  amusing  experiment  is  to  fill  the  lungs  with 
carefully  purified  hydrogen  and  repeat  deliberately  a  familiar 
speech  like  "Mary  Had  a  Little  Lamb,"  or  speak  deliberately 

12 


162  CALENDAR  OF  LEADING  EXPERIMENTS. 

the  following  words  in  order:  rude,  no,  paw,  part,  pay,  pet  and 
see,  as  explained  on  page  523,  General  Physics.  The  use  of 
familiar  words  which  are  recognizable  by  the  accompanying 
consonants  tends  to  correct  one's  perception  of  the  false  vowel 
sounds  in  spite  of  any  effort  of  the  will  on  the  listener's  part. 
Therefore  it  is  better  to  pronounce  the  successive  vowels  u,  o, 
a,  a,  a,  e  and  e,  and  point  each  time  to  a  word  (written  on  the 
blackboard)  which  contains  the  particular  vowel. 

Note. — Hydrogen  made  by  dissolving  zinc  in  sulphuric  acid 
is  apt  to  contain  arsenic  hydride.  Purify  by  passing  the  hydro- 
gen through  a  strong  solution  of  potassium  permanganate. 

153.  Illustration  of  the  action  of  the  ear  in  the  perception  of 
tone  quality  or  timbre. — Free  the  strings  of  a  piano  and  sing  a 
vowel  sound  loudly  against  the  sounding  board  as  described  and 
explained  in  Art.  373,  page  524,  General  Physics. 

154.  Beats. — Blow  two  similar  organ  pipes  together  and  alter 
the  pitch  of  one  of  the  pipes  slightly  by  means  of  a  paper  or 
cardboard  extension  of  the  open  end  of  one  of  the  pipes. 

155.  Combination    tones. — A    combination   tone    (difference 
tone)  which  can  be  heard  throughout  a  large  room  can  be  pro- 
duced as  explained  on  page  528,  General  Physics. 


PART  VI. 
•     THE   DYNAMICS  OF  WAVE  MOTION. 

The  substance  of  this  entire  part  VI  was  given  in  a  lecture  by  Wm.  S.  Franklin 
before  a  joint  meeting  of  the  Western  Society  of  Engineers  and  the  Chicago  Section 
of  the  American  Institute  of  Electrical  Engineers  on  May  28,  1917.  Incredible 
as  it  may  seem,  nearly  all  of  the  mathematics  here  given  and  all  of  the  experiments 
here  described  were  included  in  the  lecture. 


163 


SCIENCE  AND   TECHNOLOGY  VERSUS  THE  HUMANITIES  IN 
EDUCATION. 

The  worst  cant  of  our  time,  touching  the  Idolatry  of  Science*,  which  is  our  sin- 
cerest  religion,  and  handled  to  perfection  by  our  easier  college  product,  is  the  semi- 
serious  wail  of  regret  that  a  silver-spoon  smartness  was  not  transmuted  by  a  pleasant 
college  course  into  Knowledge  and  Appreciation  of  Science.  Knowledge  and 
Appreciation — alas!  It  were  better  to  say  easy  acquisition  instead  of  appreciation, 
and  fabulous  riches  instead  of  knowledge.  Riches  in  which  nearly  every  man 
has  had  a  share,  unearned. 

Unearned,  indeed,  and  yet  much  more  than  earned!  Every  loafer  knows  some- 
thing of  the  unpleasant  exactions  of  effective  labor,  but  there  never  yet  was  a 
dilettante  who  could  even  dream  of  the  pains  of  those  who  really  learn  nor  any 
self-satisfied  Philistine  who  could  sense  the  grief  of  those  who  are  wise! 

What  do  you  think  we  are  driving  at  with  all  this  apparent  rhetoric?  Merely 
to  express  the  mood  we  fall  into  when  we  read  such  things  as  Paul  Shorey's  "The 
Assault  on  Humanism"  (see  page  34  of  this  volume  for  further  comment)  and 
think  of  the  cockey  attitude  of  many  of  our  friends  who  have  to  do  with  technical 
education.  What  our  friends  need  is  to  be  obliged  to  defend  their  ground  before  a 
jury  of  Paul  Shoreys,  and  what  the  Paul  Shoreys  need  is  to  hear  us  confess  how 
little  we  are  able  to  accomplish  in  our  teaching  of  the  mathematical  sciences.  But 
much  as  we  approve  of  Paul  Shorey's  point  of  view  we  are  very  far  from  capitulating 
to  the  easy  talk  of  educationalists  concerning  the  real  aim  of  education  as  the 
development  of  personality  and  character. 

Our  greatest  comfort  as  teachers  comes  from  the  exaggerated  idea  among  those 
who  do  not  teach  (and  among  others  who  only  pretend  to  teach)  as  to  what  can 
be  accomplished  by  teaching.  That  teacher  of  philosophy,  we  know  that  he  ac- 
complishes but  little  however  bigity  his  talk.  The  manufacturer  who  prates  about 
character,  he  knows  what  he  is  talking  about  all  right,  from  his  point  of  view, 
but  his  talk  gives  us  no  distress — at  least  no  distress  concerning  our  work.  And 
we  know  that  personality  and  character  cannot  be  developed  by  sentimentality, 
however  articulate.  Science  and  technology  versus  the  humanities;  the  antithesis 
does  not  disturb  us,  we  are  at  war  with  the  Philistines! 


*  Contempt  for  science  has  been  much  in  evidence  in  England  and  America  in 
the  past,  and  of  recent  years  science  has  been  more  and  more  idolized.  With  no 
abatement  of  Contempt,  an  Idolatry  of  science  has  developed,  and  these  two  ex- 
tremes are  blended  together  in  the  same  men  I 

Science  is  Finding  Out  and  Learning  How,  but  most  men  think  of  science  in 
terms  of  results.  These  results  have  fascinated  the  crowd,  and  the  great  majority 
of  men  "have  adopted  a  scale  of  physical  values  for  everything  in  life  with  a 
consequent  neglect  of  quality  and  a  denial  of  human  value  in  everything.  We 
have  a  philosophy  of  rectangular  beatitudes  and  spherical  benevolences,  a  theology 
of  universal  indulgence,  a  jurisprudence  which  will  hang  no  rogues;  all  of  which 
means,  in  the  root,  incapacity  of  discerning  worth  and  unworth  in  anything  and 
least  of  all  in  man.  Whereas,  Nature  and  Heaven  command  us  at  our  peril,  to 
discern  worth  from  unworth  in  everything  and  most  of  all  in  man. 


164 


THE   DYNAMICS   OF   WAVE    MOTION. 

One  of  the  most  important  items  of  needed  improvement  in 
the  curriculum  of  the  engineering  school  is  the  rejuvenation 
of  the  usual,  dried-up  and  unfruitful  course  in  theoretical 
mechanics,  and  this  discussion  of  the  simplest  aspects  of  the 
dynamics  of  wave  motion  is  intended  to  show  what  can  be 
done  to  make  this  important  branch  of  mechanics  intelligible 
to  the  engineering  student.  As  Heaviside  says,  the  physics  of 
wave  motion  is  extremely  simple  and  the  mathematics  extremely 
difficult  for  the  wave  pulse,  whereas  the  physics  is  extremely 
complicated  and  the  mathematics  extremely  simple  for  the 
sine- wave  train.  Therefore  the  consideration  of  periodic  waves* 
is  almost  entirely  excluded  from  this  discussion. 

The  important  subject  of  wave-diffusion  or  wave-distortion 
is  discussed  in  an  extremely  simple  manner  in  Arts.  121-123 
(pages  222-228)  of  Franklin  and  MacNutt's  Advanced  Electricity 
and  Magnetism,  The  Macmillan  Co.,  1915. 

The  exhaustive  mathematical  treatment  of  wave-distortion 
(due  to  resistance  and  leakage  of  a  telephone  line,  for  example) 
is  based  almost  entirely  on  considerations  such  as  are  involved 

*  A  well-known  elementary  treatise  on  physics,  before  making  use  of  the  wave 
theory  of  light  "proves"  that  light  is  wave  motion,  as  follows:  Any  effect  which  is 
periodic  and  which  travels  at  finite  velocity  is  wave  motion;  light  is  periodic  and 
it  travels  at  finite  velocity;  therefore  light  is  wave  motion!  The  idea  of  the 
Catling  gun  popped  into  our  minds  when  we  first  came  across  this  argument 
and  the  effect  of  the  Catling  gun  is  periodic  and  is  propagated  at  finite  velocity  I 
In  fact  periodicity  of  waves  in  an  extended  medium  always  depends  upon  and  is 
determined  by  actions  which  take  place  in  the  system  which  produces  or  absorbs 
the  waves  (water  waves  produced  by  wind  of  a  given  velocity  constitute  an  excep- 
tion), and  waves  have,  in  general,  no  definite  velocity,  as  everyone  knows  who 
understands  what  Heaviside  calls  wave-diffusion  or  who  understands  the  difference 
between  group  velocity  and  wave  velocity  in  Helmholtz's  theory  of  dispersion.  The 
fact  is  that  the  theory  of  periodic  waves  lends  itself  easily  to  unlimited  algebraic 
formulation  and  few  men  know  anything  beyond  these  formulas  and  the  pictures  of 
sine  curves  which  underlie  theml 

165 


166 


CALENDAR  OF   LEADING  EXPERIMENTS. 


in  the  discussion  of  the  alternating-current  transmission  line  on 
pages  141-153  of  W.  S.  Franklin's  Electric  Waves,  The  Macmillan 
Co.,  1909.  This  discussion  is  unique  in  the  simplicity  with 
which  the  troublesome  boundary  conditions  are  set  forth  in 
Figs.  132,  140-147. 

The  equation  of  a  traveling  curve. — The  curve  cc,  Fig.  72,  is 
stationary  with  respect  to  the  origin  of  coordinates  0',  and  the 
curve  and  the  origin  0'  are  assumed  to  be  traveling  together 
towards  the  right  at  velocity  v  so  that  the  abscissa  of  the 


-•stationary 

y-axis 

Amoving 
r  y-axis 

X"v 

> 

/      > 

^ 

-j^/ 

V 

ct 

x^axis 

t  et- 

.  JC 

'  >i 
,1 

I 

Tig.  72. 

moving  origin  0'  as  referred  to  the  fixed  origin  O  is  at  each 
instant  equal  to  vt  as  indicated  in  the  figure.  Therefore  the 
abscissa  x  of  the  point  p  on  the  moving  curve  is  x  =  x'  -f-  vt, 
so  that  x'  =  x  —  vt.  Let  the  equation  to  the  moving  curve  cc 
as  referred  to  the  moving  origin  0'  be 


y  = 


Then,  substituting  x  —  vt  for  x't  we  have 

y  =  F(x-  vt) 


(i) 


as  the  equation  of  a  curve  traveling  to  the  right  at  velocity    v; 
and  in  a  similar  manner  it  may  be  shown  that 


y  =f(x  +  vt) 


(2) 


is  the  equation  of  a  curve  'traveling  to  the  left  at  velocity   v. 
It  is  highly  instructive  to  derive  a  differential  equation  which 


THE  DYNAMICS  OF  WAVE  MOTION.  167 

is  satisfied  by  both  equations  (i)  and  (2);  and,  inasmuch  as  the 
rules  for  differentiating  are  not  usually  understood  by  the  student, 
especially  when  applied  to  such  general  equations  as  (i)  and  (2), 
the  following  discussion  is  arranged  to  appeal  to  one's  funda- 
mental arithmetical  sense.  Awkwardness  of  notation  is  the 
chief  difficulty,  as  usual,  and  one  must  remember  the  following 
points : 

If  y  increases  always  u  times  as  fast  as  x,  then  u  is  called 
the  derivative  of  y  with  respect  to  x.  If  y  depends  on  x  as 
the  only  variable,  the  derivative  is  represented  by  the  symbol 
dy/dx.  If  y  depends  on  more  than  one  variable,  the  derivative 
with  respect  to  x  is  called  a  partial  derivative  and  it  is  usually 
represented  by  the  symbol  dy/dx.  The  two  symbols  dy/dx 
and  dy/dx  have,  however,  precisely  the  same  arithmetical  mean- 
ing; thus  in  the  first  case  y  actually  increases  dy/dx  times 
as  fast  as  x,  and  in  the  second  case  y  would  increase  dy/dx 
times  as  fast  as  x  if  x  only,  among  all  the  variables  upon 
which  y  depends,  were  allowed  to  change. 

If  u  (=  dy/dx  or  dy/dx)  increases  w  times  as  fast  as  x, 
then,  of  course,  w  is  the  derivative  of  u  with  respect  to  x  and 
of  course  it  is  represented  by  du/dx  or  du/dx.  In  thinking  of  the 
derivative  of  u,  however,  it  is  often  desirable  to  keep  clearly 
in  mind  the  fact  that  u  itself  is  the  derivative  of  y  with  respect 
to  x.  This  is  shown  by  the  following  notation: 

du      d*y  dy 

—  =  —     when     u  =  3- 
dx       dx2  dx 

du       d2y  dy 

T~  =  Tl     when     u  =  — 
dx       dx2  dx 

Partial  differential  equation  of  travel. — The  two  equations  (i) 
and  (2),  above,  are  particular  solutions  of  a  partial  differential 
equation  which  may  be  called  the  partial  differential  equation  of 
travel,  namely 

dzy          d2y 

* +  (3) 


168  CALENDAR  OF  LEADING  EXPERIMENTS. 

and  this  differential  equation  is  of  fundamental  importance  in 
the  mathematical  theory  of  wave  motion. 

For  the  sake  of  simplicity  let  the  quantity  (x  —  vt)  of  equa- 
tion (i)  be  represented  by  z  so  that 

z  =  x  —  vt  (i) 

from  which  we  have  dz/dx  =  i  (which  means  that  if  x,  only, 
increases  z  must  increase  at  the  same  rate),  and  dz/dt  =  —  v 
(which  means  that  if  /,  only,  increases  z  must  increase  —  v 
times  as  fast  or  decrease  v  times  as  fast). 

Equation  (i)  is  to  be  thought  of  for  the  moment  as  y  —  F(z), 
and  the  derivative  dy/dz  may  be  represented  by  F'(z),  meaning 
that  y  increases  F'(z)  times  as  fast  as  z. 

Proposition:  y  increases  F'(z)  times  as  fast  as  z,  and  when 
x,  only,  changes  z  increases  dz/dx  times  as  fast  as  x.  Therefore 
when  x,  only,  changes  y  must  increase  F'(z)  times  dz/dx 
times  as  fast  as  x.  Therefore  dy/dx  =  F'(z)>  (dz/dx).  But 
dz/dx  =  i  as  stated  above.  Therefore* 


Proposition:  y  increases  F'(z)  times  as  fast  as  z,  and  when 
/,  only,  changes  z  increases  dz/dt  times  as  fast  as  /.  Therefore 
when  /,  only,  changes  y  must  increase  F'(z)  times  dz/dt  times 
as  fast  as  /.  Therefore  dy/dt  =  F'(z)-  (dz/dt).  But  dz/dt  =  -  v 
as  stated  above.  Therefore 

g=  -v-F'(z)  (iii) 

From  equations  (ii)  and  (iii)  it  is  evident  that  dy/dt  = 
—  v  (dy/dx),  and  this  may  be  called  the  differential  equation 
of  travel  to  the  right.  If  we  had  started  with  y  =  f(x  +  vt) 

*  To  the  student  who  is  familiar  with  the  rules  for  partial  differentiation  this 
argument  may  seem  ridiculous,  but  it  is  not  ridiculous,  by  any  means.  No  one 
can  understand  partial  differentiation  who  has  not  at  some  time  followed  this 
kind  of  an  argument  in  pure  arithmetic. 


THE   DYNAMICS  OF  WAVE   MOTION.  169 

and  z  =  x  -f-  ^»  the  above  discussion  would  have  led  to 
dy/dt  =  +  v-  (dy/dx),  which  may  be  called  the  differential  equa- 
tion of  travel  to  the  left.  Neither  of  these  differential  equations 
is  of  importance  in  the  theory  of  wave  motion. 

Let  us  think  of  the  function  F'(z)  as  changing  F"(z)  times 
as  fast  as  z. 

When  x  alone,  changes,  z  changes  dz/dx  times  as  fast  as  x, 
and  therefore  F'(z)  changes  F"(z)  times  dz/dx  times  as  fast 
as  x.  But  F'(z)  is  equal  to  dy/dx  according  to  equation  (ii). 
Therefore  dy/dx  increases  F"(z)  times  dz/dx  times  as  fast  as  x. 
Therefore,  since  dz/dx  =  I,  we  have 


When  /  alone,  changes,  z  changes  dz/dt  times  as  fast  as  t\ 
therefore  F'(z)  increases  F"(z)  times  dz/dt  times  as  fast  as  t,  and 
therefore  —v-F'(z)  increases  —  v  times  F"(z)  times  dz/dt  times 
as  fast  as  t.  But  —  v-F'(z)  is  equal  to  dy/dt  according  to 
equation  (iii).  Therefore  dy/dt  increases  -  v  X  F"(z)  X  dz/dt 
times  as  fast  as  /,  and  therefore,  since  dz/dt  =  —  v,  we  have 

fjr  =  *F'(?)  (v) 

From  equations  (iv)  and  (v)  we  have 


and  this  same  result  would  be  obtained  from  equation  (2)  by 
taking  z  =  x  +  vt.  That  is  to  say,  equation  (3)  is  a  differential 
equation  which  is  satisfied  by  both  of  the  equations  (i)  and  (2), 
and  it  is  of  very  great  importance  in  the  mathematical  theory 
of  wave  motion. 

The  principle  of  superposition.  A  property  of  linear  differ- 
ential equations.  —  A  principle  of  extremely  wide  application  in 
physics  is  the  so-called  principle  of  superposition.  From  the 


170  CALENDAR  OF   LEADING   EXPERIMENTS. 

physical  point  of  view  a  general  statement  of  the  principle  is 
scarcely  possible,  and  therefore  the  following  examples  must 
suffice:  (i)  A  person  at  A  in  Fig.  73  can  see  window  No.  i 

and  another  person  at  B 
can  see  window  No.  2  at 
the  same  time.  This  means 
that  the  two  beams  of  light 
a  and  b  can  travel  through 
the  same  region  at  the  same 
time  without  getting  tangled 

Fig  73  together,    as    it    were;    each 

beam   behaves  as   if   it  were 

traveling  through  the  region  alone.  (2)  Two  sounds  can  travel 
through  the  same  body  of  air  simultaneously,  and  each  sound 
travels  as  though  it  occupied  the  space  by  itself.  (3)  Two 
systems  of  water  waves  can  travel  over  the  same  part  of  a  pond 
simultaneously,  each  system  behaving  as  if  the  other  were  not 
present.  (4)  Two  messages*  can  travel  over  a  telegraph  wire 
simultaneously  and  not  get  mixed  up  together.  (5)  Two  forces 
F  and  G  exerted  simultaneously  upon  an  elastic  structure 
produce  an  effect  which  is  the  sum  of  the  effects  which  would  be 
produced  by  the  forces  separately,  provided  the  sum  of  the  forces 
does  not  exceed  the  elastic  limit  of  the  structure,  therefore  each 
force  may  be  thought  of  as  producing  the  same  effect  that  it 
would  produce  if  acting  alone. 

All  of  the  effects  in  physics  which  are  superposable — and  this 
includes  by  far  the  greater  portion  of  the  effects  in  mechanics, 
heat,  electricity  and  magnetism,  light  and  sound,  and  chemistry 
— are  expressible  in  terms  of  linear  differential  equations,  and 
the  principle  of  superposition  is  a  clearly  defined  property  of 
such  equations  as  follows:  //  y  is  a  function  of  x  which  satisfies 

*  Indeed  any  number  of  distinct  messages  can  travel  over  a  telegraph  wire 
in  either  direction  or  in  both  directions  simultaneously.  The  only  limiting  feature 
in  multiplex  telegraphy  is  the  design  of  the  sending  and  receiving  apparatus; 
and  the  same  is  true  in  wireless  telegraphy.  In  each  of  the  above  examples  the 
word  two  means  two  or  more. 


THE   DYNAMICS  OF  WAVE  MOTION.  lyi 

a  linear  differential  equation,  and  if  z  is  another  function  of  x 
which  satisfies  the  same  differential  equation ,  then  (y  +  z)  is  a 
function  which  satisfies  the  differential  equation. 

This  proposition  is  true  for  both  ordinary  and  partial  linear 
differential  equations,  and  indeed  nearly  all  of  the  superposable 
effects  in  physics  are  expressible  in  terms  of  partial  linear  dif- 
ferential equations.  The  proof  of  the  proposition  is,  however, 
nearly  the  same  for  ordinary  and  for  partial  differential  equations, 
and  therefore  it  is  sufficient  to  give  the  proof  for  ordinary 
differential  equations  only.  Let  the  given  linear  differential 
equation  be: 


Let  y  be  a  function  of  x  which  satisfies  this  differential  equation, 
then: 

Let   z  be  another  function  of  x  which  satisfies  (i),  then: 


Now 


2)  _  dy      dz  dz(y 

~       + 


dx        ~  dx      dx  dx2        ~  dxz      dx* 

Therefore,  adding  equations  (ii)  and  (iii)  ,  we  get  : 

4-  z) 


But  equation  (iv)  is  exactly  the  same  form  as  equation  (i), 
and  therefore  (y  +  z)  is  a  function  of  x  which  satisfies  equation 

(o.* 

*  The  above  proposition  is  true  for  any  linear  differential  equation  whatever; 
that  is  when  the  coefficients  A,  B,  etc.,  are  constants,  and  when  the  coefficients 
A,  B,  etc.,  are  functions  of  the  independent  variable  x.  The  latter  type  of  linear 
differential  equation  does  not,  however,  concern  us  here. 


172  CALENDAR  OF  LEADING  EXPERIMENTS. 

The  above  proposition  is  the  basis  of  Fourier's  method  of 
analysis  as  applied  to  the  flow  of  heat  and  as  applied  to  the 
motion  of  strings,  and  it  is  the  basis  of  the  use  of  spherical, 
zonal,  and  cylindrical  harmonics.  The  importance  of  the  propo- 
sition can  scarcely  be  overestimated. 

Undetermined  constants  in  the  solution  of  an  ordinary  differ- 
ential equation.  —  It  is  sufficient,  perhaps,  to  illustrate  this  matter 
by  two  very  simple  examples. 

Example  I.  —  Consider  the  simple  ordinary  differential  equation 

• 

dy 


where    a    is  a  given  constant.     The  increase  of    y    during    / 
seconds  is  at,  and  the  value  of  y  at  the  end  of  the  /  seconds  is 

y  =  at  +  c 

where    C  is  the  unknown  value  of   y  at  the  beginning    (/  =  o). 
Example  2.  —  Consider  the  simple  ordinary  differential  equation 


where  a   is  a  given  constant.     Then 


and 

+  Bt  +  C 


where    C  is  the  unknown  value  of   y   at  the  beginning,  and   B 
is  the  unknown  value  of  dyjdt  at  the  beginning. 

Note.  —  From  the  point  of  view  of  the  physicist  the  unknown 
constants  which  appear  in  the  general  solution  of  an  ordinary 
differential  equation  are  thought  of  as  disposable  constants  because 
the  solution  may  be  made  to  fit  any  special  case  by  assigning 
proper  values  to  these  constants.  The  number  of  disposable 


THE   DYNAMICS  OF  WAVE   MOTION. 


173 


constants  in   the  general  solution  of  an  ordinary  differential 
equation  is  always  equal  to  the  order  of  the  differential  equation. 

Undetermined  functions  in  the  solution  of  a  partial  differential 
equation. — It  is  sufficient,  perhaps,  to  illustrate  this  matter  by 
a  few  simple  examples. 

Example  I. — Concerning  a  hill  it  is  known  only  that  its  slope, 
dy/dx,  in  the  direction  of  the  #-axis  is  constant  and  equal  to  a 
so  that 


(0 


Before  the  complete  hill  or  surface  can  be  constructed  from 
this  differential  equation  an  arbitrary  starting  curve  cc,  Fig.  74, 
must  be  chosen.  Let  y  =  F(z)  be  the  equation  to  the  curve  cc, 
then  F(z)  is  the  height  of  the 


hill  at  any  point  z  on  cc,  and 
ax  +  F(z)  is  the  height  of  the  hill 
at  any  point  x,  z.  That  is  to 
say,  the  integration  of  (i)  gives 


w-axis 


y  =  ax  +  F(z) 


(ii) 


Fig.  74. 


Example  2.  —  If  all  boys  were  of 
the  same  ability  we  might  say 
that  any  boy  saves  money  at 
the  average  rate  of  $5  per  year 
beginning  at  fourteen  years  of  age.  Integrating  with  respect 
to  b,  the  boy,  from  fourteen  years  to  twenty-one  years  we  get 
$35;  but  the  amount  of  money  a  boy  has  when  he  comes  of 
age  is  not  $35  plus  a  constant,  but  $35  +  F(m),  where  F(m) 
is  what  the  boy's  "old  man"  has  saved  for  him;  b  and  m  are 
independent  variables,  let  us  say,  and  a  "constant"  of  integra- 
tion with  respect  to  b  turns  out  to  be  an  unknown  function  of  m. 

Example  j.  —  Concerning  a  hill  it  is  known  only  that  its  slope, 
dy/dx,    in  the  direction  of  the  #  -axis  increases    a    times  as  fast 


174  CALENDAR  OF  LEADING  EXPERIMENTS. 

as  xt   or,  expressed  in  symbols,  we  have 


Before  the  complete  hill  or  surface  can  be  constructed  from  this 
differential  equation  two  things  must  be  given,  namely,  (i)  An 
arbitrary  starting  curve  like  cc,  Fig.  74,  and  (2)  An  arbitrary 
value  of  the  starting  slope,  dy/dx,  at  each  point  of  cc. 

The  integration  of  (i)  or  (iii)  is  called  partial  integration,  but 
example  2  should  make  it  clear  that  partial  integration  is  identical 
to  ordinary  integration,  the  only  difference  being  that  in  the 
former  the  "constant"  of  integration  turns  out  to  be  a  function 
of  the  other  independent  variable  or  variables.  Therefore, 
integrating  (iii)  twice  we  gee 

y  =  J«*  +  *•/(«)  +  F(z)  (iv) 

where  f(z)  and  F(z)  are  unknown  functions  of  z.  Indeed 
y  =  F(z)  is  the  equation  to  the  starting  curve  cc  in  Fig.  74, 
and  the  value  of  the  starting  slope  (dy/dx)x=0,  at  each  point  of 
cc  is  (dy/dx)x=0,  =/(*). 

Note.  —  From  the  point  of  view  of  the  physicist  the  unknown 
functions  which  appear  in  the  general  solution  of  a  partial 
differential  equation  are  thought  of  as  disposable  functions 
because  the  solution  may  be  made  to  fit  any  special  case  by 
properly  choosing  the  forms  of  these  functions.  The  number  of 
these  disposable  functions  in  the  general  solution  of  a  partial 
differential  equation  is  always  equal  to  the  order  of  the  differential 
equation. 

General  solution  of  equation  (3).  —  Equations  (i)  and  (2)  are 
particular  solutions  of  (3),  and  therefore 

y  =  F(x-vt)  +  /(*  +  »/)  (4) 

is  also  a  solution,  according  to  the  principle  of  superposition. 
But  equation  (4)  involves  two  independent  and  disposable  func- 
tions and  it  is  therefore  the  general  solution  of  (3). 


THE   DYNAMICS  OF   WAVE   MOTION. 


175 


Equation  (4)  represents  the  piling  on  top  of  each  other  of  two 
curves  of  any  shape,  one  of  the  curves  traveling  to  the  right  at 
velocity  v,  and  the  other  traveling  to  the  left  at  velocity  v. 

Differential  equation  of  motion  of  a  stretched  string. — When 
a  stretched  string  is  in  equilibrium  it  is  of  course  straight.  Let 
us  choose  this  equilibrium  position  of  the  string  as  the  #-axis 
of  reference.  We  will  assume  that  each  particle  of  the  string 
moves  only  in  a  direction  at  right  angles  to  the  string  (parallel 
to  the  j-axis  of  reference),  and  we  will  assume  that  the  string 
is  perfectly  flexible  which  means  that  the  only  forces  to  be  con- 
sidered are  the  forces  due  to  the  tension  of  the  string.  An 
important  consequence  of  the  first  assumption  is  that  the  x- 
component  of  the  tension  of  the  string  has  always  and  every- 
where a  certain  value  T  which  is  equal  to  the  tension  of  the 
string  when  it  is  in  equilibrium. 


Fig.  75. 


Let  the  curve  ccc,  Fig.  75,  be  the  configuration  of  the  string 
at  a  certain  instant,  that  is,  ccc  is  what  the  photographer  would 
call  a  snapshot  of  the  moving  string.  The  shape  of  the  curve 
ccc  defines  y  as  a  function  of  x  and  the  steepness  of  the  curve 
at  any  point  is  the  value  of  dy/dx*  at  that  point. 

Consider  the  very  short  portion  ab  of  the  string.  The  length 
of  this  portion  when  the  string  lies  along  the  #-axis  (in  equi- 
librium) is  dxj  and  the  mass  of  the  portion  is  m-dx,  where  m 
is  the  mass  per  unit  length  of  string.  An  enlarged  view  of  the 
very  short  portion  ab  of  the  string  is  shown  in  Fig.  76.  The 

*  Any  reader  who  fails  to  appreciate  the  fact  that  we  are  here  talking  about  the 
state  of  affairs  at  a  given  instant,  or  that  time  is  supposed  to  stop,  as  it  were, 
may  wonder  why  we  use  the  notation  dy/dx  instead  of  dy/dx. 


176  CALENDAR  OF  LEADING  EXPERIMENTS. 

adjacent  portions  of  the  string  pull  on  the  portion  ab,  the  pull 
at  a  is  represented  by  R  and  it  is  parallel  to  the  string  at  a, 
and  the  pull  at  b  is  represented  by  R'  and  it  is  parallel  to  the 
string  at  b.  The  ^-component  of  R  is  the  force  T  to  the  left, 
and  the  ^-component  of  R'  is  an  equal  force  T  towards  the 
right.  Therefore  the  downward  force  D  (see  Fig.  76)  is  equal 
to  T  tan  6,  the  upward  force  U  is  equal  to  T  tan  B'  ',  and  the 
net  upward  force  acting  on  the  portion  ab  of  the  string  is  : 

dF  =  U  -  D  =  T  tan  tf  -  T  tan  0  (i) 

But  tan  B  is  equal  to  the  value  of  dy/dx  at  a,  and  tan  6'  is 
equal  to  the  value  of  dy/dx  at  b.  Therefore  the  value  of 
tan  B'  —  tan  B  is  the  increase  of  dy/dx  from  a  to  6,  and  this 
increase  is  equal  to  (d2y/dx2)  -dx.  This  is  evident  when  we  con- 
sider that  d2y/dx2  means  the  rate  of  increase  of  dy/dx  with 
respect  to  x.  Therefore,  substituting  (d^/dx^'dx  for 
tan  0'  —  tan  B  in  equation  (i)  we  get: 


Now,  according  to  Newton's  laws  of  motion,  the  net  upward 
force  dF  acting  on  the  portion  ab  of  the  string  is  equal  to  the 
mass  m-dx  of  the  portion  multiplied  by  the  upward  acceleration, 
Bzy/dP,  of  the  portion.  Therefore  substituting  m(d2y/dP)-dx 
for  dF  in  equation  (ii)  we  get  : 

d*y          d*y 

m~W  ~  TM 
or 

dz  T    d* 


The  general  solution  of  this  differential  equation,  as  explained 
above,  is: 

y  =  F(x  -  vf)  +  f(x  +  vt)  (4)  bis 


THE   DYNAMICS  OF   WAVE   MOTION.  177 

where   v   is  given  by  the  equation : 


.-£ 


Of  course  particular  solutions  of  equation  (3)  are 

y  =  F(x  -  vt)  (6) 

and 

y=f(x  +  vt)  (7) 

Equation  (6)  represents  a  bend  or  curve  on  the  string  traveling 
to  the  right  at  velocity  v  and  retaining  its  shape  unchanged, 
and  equation  (7)  represents  a  bend  or  curve  on  the  string  traveling 
to  the  left  at  velocity  v  and  retaining  its  shape  unchanged. 
These  traveling  bends  of  unchanging  shape  are  called  pure  waves. 

Pure  wa've  traveling  to  the  right. — Equation  (6)  expresses 
what  is  called  a  pure  wave  traveling  to  the  right  and  it  is  important 
to  consider  the  necessary  relation  between  V  and  s  where  V 
is  the  sidewise  velocity  and  5  is  the  slope  of  the  string  at  a  point ; 
of  course,  V  =  dy/dt  and  s  =  dy/dx. 

From  equation  (6)  we  have  s  =  (dy/dx)  =  F'(x  —  vt),  and 
V  =  dy/dt  =  —  v-F'(x  —  vt),  and  therefore  for  a  pure  wave 
traveling  to  the  right  we  have 

.     -  ;       r   7  =  -«  ,;      (8) 

Pure  wave  traveling  to  the  left. — From  equation  (7)  we  have 

^  =  dy/dx  =  F'(x  +  vt)  and  V  =  dy/dt  =  +  v-F'(x  +  v),  and 
therefore  for  a  pure  wave  traveling  to  the  left  we  have 

j=+v  (9) 

Remark. — Equations  (8)  and  (9)  are  useful  in  that  they  enable 
one  to  pick  out  particular  solutions  of  equation  (3).     Any  dis- 
tribution of  sidewise  velocity  V  and  slope  5  which  satisfies  (8) 
is  a  pure  wave  traveling  to  the  right,  and  any  distribution  of 
13 


178  CALENDAR  OF  LEADING   EXPERIMENTS. 

sidewise  velocity  V  and  slope  5  which  satisfies  (9)  is  a  pure  wave 
traveling  to  the  left. 

Reflection  and  change  of  phase  thereby.  —  Let  us  consider  a 
simple  form  of  pure  wave  traveling  to  the  right  along  a  string, 
a  wave  throughout  which  the  sidewise  velocity  of  the  string  has 
everywhere  the  same  value  V,  and  throughout  which  the  slope 
of  the  string  has  everywhere  the  same  value  s.  Such  a  simple 
wave  we  will  call  a  ribbon  wave,  and  it  may  be  symbolized  by  the 
arrow  in  Fig.  77.  The  head  of  the  arrow  shows  the  direction  of 
travel,  everywhere  between  the  points  a  and  b  the  string  is 
moving  sidewise  at  velocity  V,  and  everywhere  between  a 
and  b  the  slope  of  the  string  is  s.  This  slope  is  not  actually 
shown  however. 

What  happens  when  the  wave  V  s  reaches  the  end  of  the 
string?  Whatever  happens  on  the  string  it  must  be  of  the  nature 
of  a  pure  wave  traveling  to  the  right  and  a  pure  wave  traveling 
to  the  left,  one  heaped  on  top  of  the  other,  because  this  is  the 
physical  interpretation  of  the  general  equation  (4).  Therefore  if 
we  assume  a  pure  wave  (sidewise  velocity  V  and  slope  sf)  to 

;o         string  fc          E  string  <Y*    *' 


Fig.  77.  Fig.  78. 

Reflection  from  rigid  end  of  string. 

shoot  to  the  left  from  E  when  ab  reaches  E  we  are  sure  to 
cover  every  possible  outcome,  and  then  if  this  assumption  is 
justified  by  showing  that  it  satisfies  all  of  the  necessary  conditions 
we  may  be  sure  that  the  correct  solution  has  been  found.  In 
Fig.  78,  therefore,  is  shown  the  tail  end  of  the  original  wave  V  s, 
and  the  beginning  of  an  assumed  reflected  wave  V  s'.  Now, 
according  to  equations  (8)  and  (9)  we  must  have 


and 

V 


THE   DYNAMICS  OF   WAVE   MOTION.  179 

Furthermore  the  actual  end  of  the  string  cannot  move  and 
therefore  we  must  have 

V  +  V  =  o  (iii) 

and  from  equations  (i),  (ii)  and  (iii)  it  follows  that  V  =  —  V 
and  that  s'  =  s.  That  is  to  say,  a  wave  is  wholly  reflected  from 
the  rigid  end  of  a  string  (numerical  values  of  V  and  V  the 
same,  and  numerical  values  of  s  and  sf  the  same),  and  the  side- 
wise  velocity  of  the  string  in  the  reflected  wave  is  the  reverse  of 
the  sidewise  velocity  of  the  string  in  the  original  wave 
(V  •=  -  V). 

Figure  79  represents  an  ideal  condition  in  which  the  end  of  a 
string  is  held  by  an  indefinitely  long  weightless  thread.  In  this 
case  the  actual  end  of  the  string  cannot  slope,  and  therefore  we 

must  have 

5  +  sf  =  o  (iv) 

which  in  conjunction  with  equations  (i)  and  (ii)  gives  s'  =  —  s 


string                 <-*        weightless 
•^ — i thread 

Fig.  79.  Fig.  80. 

Reflection  from  "free"  end  of  string. 

and  V  =  V.  That  is,  a  wave  is  wholly  reflected  (numerical 
values  of  5  and  s'  the  same,  and  numerical  values  of  V  and 
V1  the  same)  at  the  "free"  end  of  a  stretched  string,  and  the 
slope  in  the  reflected  wave  is  opposite  to  the  slope  in  the  original 
wave  (sf  =  —  s).  The  "free"  end  of  a  stretched  string  as 
shown  in  Fig.  79  is  merely  an  ideal,  but  the  analogous  condition 
in  electric  waves  and  in  air  waves  is  very  common. 

Examples  of  wave  motion  on  a  string. — i .  A  complete  picture 
of  a  ribbon  wave  traveling  to  the  right  on  a  stretched  string  is 
shown  in  Fig.  80.  The  region  ab  occupied  by  the  wave  has  a 
uniform  slope  5,  and  every  part  of  the  string  ab  is  moving  down- 
wards at  velocity  V  as  indicated  by  the  small  parallel  arrows. 


ISO  CALENDAR  OF   LEADING   EXPERIMENTS. 

2.  An  indefinitely  long  stretched  string  is  struck  sharply  by  a 
square-faced  hammer  so  as  to  set  every  particle  of  the  portion  ab 
of  the  string  moving  sidewise  at  a  given  velocity  V  as  indicated 
by  the  small  parallel  arrows  in  Fig.  81.  To  determine  the  motion 
of  the  string  it  is  only  necessary  to  resolve  the  uniform  sidewise 


V       -« 
string  B^   V        +s 


•llUUlr 

Fig.  82. 

velocity  and  zero  slope  of  portion  ab  into  two  oppositely  traveling 
pure  waves  as  indicated  in  Fig.  82,  in  which  the  arrow  A  repre- 
sents a  ribbon  wave  traveling  to  the  right  and  the  arrow  B 
represents  a  ribbon  wave  traveling  to  the  left.  Half  of  the  given 
sidewise  velocity  is  associated  with  each  ribbon  wave  as  indicated, 
and  equal  and  opposite  slopes  [according  to  equations  (8)  and  (9)] 
are  associated  with  the  respective  ribbon  waves.  The  state  of 
affairs  a  moment  later  is  shown  in  Fig.  83,  and  Fig.  84  shows  the 


Fig.  83.  Fig.  84. 

state  of  affairs  after  the  two  ribbon  waves  A  and  B  have 
entirely  separated  from  each  other. 

3.  Motion  of  a  plucked  string. — A  stretched  string  is  pulled 
sidewise  into  the  position  APB,  Fig.  85,  and  released;  and  it 
is  required  to  determine  the  motion  of  the  string.  For  the  sake 
of  simplicity  the  point  P  (or  C)  is  taken  at  the  middle  of  the 
string. 

To  determine  the  motion  of  the  string  the  initial  condition 
of  the  string,  namely,  slope  +  a  between  A  and  C,  and 


THE   DYNAMICS  OF  WAVE   MOTION. 


slope  —  a  between  C  and  B,  with  no  sidewise  velocity  any- 
where, must  be  resolved  into  pure  waves  traveling  to  right  and 
left.  Thus  the  slope  —  a  is  resolved  into  the  two  waves  s'  V 
and  sn  V",  and  the  slope  +  Q>  is  resolved  into  the  two  waves 
s'"  V"  and  siy  FIV. 

Now    V   must  be  equal  to    —  V"   because  there  is  zero  side- 
wise  velocity  between    C  and   B,    therefore,  according  to  equa- 


Fig.  85. 


tions  (8)  and  (9),  s'  must  be  equal  to  s"  and  the  same  in  sign. 
But  s'  +  s"  =  -a  so  that  s'  =  s"  =  -  a/2. 

Similarly,  V"  must  be  equal  to  —  FIV,  so  that  s'"  must  be 
equal  to  5IV  and  the  same  in  sign.  But  S"'  -f  5IV  =  +  a 
so  that  S'"  =  5IV  =  +  a/2. 

Therefore,  using  a/2  for  the  common  numerical  value  of 
s',  s",  s'"  and  $IV,  and  using  V  for  the  common  numerical 
value  of  F',  F",  V"  and  FIV,  and  indicating  the  proper  sign 
in  each  case  we  may  simplify  Fig.  85  as  indicated  in  Fig.  86. 

To  find  the  state  of  affairs  after  lapse  of  time  t  it  is  only  neces- 
sary to  consider  that  the  ribbon  waves  will  all  have  traveled 
forwards  a  distance  vt,  and  that  reflection  at  A  and  B  always 
takes  place  with  reversal  of  F.  Thus  after  time  sufficient  for 
waves  to  travel  over  ij  times  the  length  of  the  string  (arrow- 
head p,  for  example,  will  have  traveled  to  A  and  back  to  q, 


_ 

1  1  1  1  t*Ft  t  1  1  1  r 


+V 


moving 


C 


-40     - 


•TV 


Fig.  87 


182 


CALENDAR  OF  LEADING   EXPERIMENTS. 


and  —  V  will  have  been  converted  into  -f-  V  by  reflection  at 
A)  the  state  of  affairs  will  be  as  shown  in  Fig.  87.  A  portion 
of  the  string  near  A  is  still  and  its  slope  is  —  a;  the  middle 
portion  of  the  string  is  moving  upwards  at  velocity  2V  and 
its  slope  is  zero;  and  the  portion  near  B  is  still  and  its  slope  is 
+  O"  Slope  and  velocity  of  each  part  of  string  in  Fig.  87  is  found 
by  adding  slopes  and  velocities  associated  with  the  overlapping 
portions  of  the  respective  ribbon  waves.* 

The  Kelvin  ladder. — A  number  of  equidistant  bars  are  fixed 
to  a  fine  steel  wire  or  ribbon  and  suspended  from  a  small  crank 
as  indicated  in  Fig.  88.  The  rotatory  motion  of  each  portion  of 
this  arrangement  satisfies  the  equation 


E 

dt?  ^  k      dx2 

where  <f>  is  the  angular  displacement  at  the  instant  /  of  the 
bar  which  is  at  a  distance  x  below  the  upper  end  of  the  ladder 
(jc-axis  directed  downwards),  k  is  the  moment; 
inertia  of  unit  length  of  the  ladder,  and  E  is  the 
torque  required  to  produce  one  radian  of  twist  per 
unit  length  of  the  ladder.  This  equation  may  be 
established  by  an  argument  which  is  nearly  identical 
in  form  to  the  argument  leading  to  equation  (3) 
on  page  176,  and  what  we  may  call  a  ribbon  wave 
may  be  symbolized  by  the  arrow  in  Fig.  88.  Every 
portion  of  the  ladder  between  the  points  a  and 
b  is  rotating  at  uniform  spin-velocity,  and  the 
whole  of  the  portion  ab  of  "the  ladder  is  uniformly 
twisted.  Let  o>  be  the  spin-velocity  and  h  the 
degree  of  twist  (radians  per  unit  length  of  ladder). 
Then  the  ratio  u/h  has  a  certain  value,  a,  which 
is  positive  or  negative  according  to  whether  the 
wave  is  traveling  upwards  or  downwards. 

*  A  more  elaborate  example  of  the  motion  of  a  plucked  string  as  analyzed  by 
this  same  method  is  given  in  Journal  of  the  Franklin  Institute,  Vol.  CLXXIX, 
May,  1915. 


Fig.  88. 


THE   DYNAMICS  OF   WAVE   MOTION. 


183 


When  a  ribbon  wave  is  reflected  at  the  free  end  B  of  the  ladder, 
the  twist  h  is  reversed  without  change  of  sign  of  co.  That  is 
to  say,  if  the  ladder  is  twisted  like  a  right-handed  screw  or  helix 
in  the  original  wave  it  will  be  twisted  as  a  left-handed  screw  or 
helix  in  the  reflected  wave,  but  the  rotation  or  spin  in  the 
reflected  wave,  will  be  in  the  same  direction  as  the  rotation  or 
spin  in  the  original  wave. 

When  the  end  bar  at  B  is  rigidly  fixed,  a  ribbon  wave  is 
reflected  at  B  with  reversal  of  rotation  or  spin  but  without 
reversal  of  twist. 

What  takes  place  when  an  elastic  rod  strikes  endwise  against 
a  rigid  wall  and  rebounds. — A  longitudinal  wave  on  a  rod  is  a 
state  of  endwise  compression  C  (negative  value  of  C  means 
tension)  associated  with  a  certain  velocity  V  of  the  material  of 
the  rod,  and  the  ratio  V/C  has  a  certain  value,  a,  which  is 
positive  or  negative  according  to  the  direction  of  travel  of  the 
wave.  The  initial  condition  of  the  elastic  rod  just  before  it 
strikes  the  wall  is  a  uniform  velocity  of  the  material  of  the  rod 
towards  the  wall,  this  initial  condition  is  to  be  thought  of  as 
always  being  in  existence  except  in  so  far  as  it  is  modified  by 
the  superposition  of  new  conditions,  and  this  initial  condition 
of  uniform  unchanging  motion  is  symbolized  by  the  heavy  dotted 
line  in  Figs.  89  and  90. 


V      +C 


Fig.  89. 


B       moving               still 

••.'••  •.'/•'.•'.*•:•:'•':  '* 

not  compressed 

compressed 

[     -V     +C 

-V      -C 

Fig.  90. 


After  the  rod  strikes  the  wall  it  continues  for  a  definite  time 
to  push  steadily  against  the  wall,  a  long-drawn-out  ribbon  wave 
continues  to  shoot  out  from  the  wall  as  indicated  by  the  heavy- 


184  CALENDAR  OF  LEADING   EXPERIMENTS. 

line  arrow  in  Figs.  89  and  90.  The  first  lap  of  this  ribbon  wave 
(velocity  —  V  of  the  material  of  the  rod  and  an  associated 
compression  C)  annuls  or  literally  wipes  out  the  initial  velocity 
V  of  the  rod,  and  lays  down  a  condition  of  uniform  compression 
as  indicated  in  Fig.  89.  The  second  lap  of  the  ribbon  wave 
(velocity  —  V  and  compression  —  C,  because  reflection  at 
the  free  end  B  of  the  rod  takes  place  with  reversal  of  C  so 
that  the  compression  in  the  first  lap  becomes  tension  in  the  second 
lap)  then  wipes  out  the  uniform  compression  and  lays  down  a 
uniform  condition  of  reversed  motion  as  indicated  in  Fig.  90. 
Therefore  when  the  second  lap  of  the  ribbon  wave  reaches  the 
wall  the  entire  rod  is  relieved  of  compression  and  it  is  moving 
away  from  the  wall  at  velocity  V. 

Sudden  stopping  of  flow  of  water  in  a  rigid  pipe. — When  a  valve 
is  suddenly  closed  at  the  end  of  a  pipe  the  stoppage  and  re- 
bounding of  the  column  of  moving  water  takes  place  precisely 
in  the  manner  of  the  stoppage  and  rebounding  of  the  rod  as 
represented  in  Figs.  89  and  90,  on  the  assumption  that  the  pipe 
is  rigid. 

Experiment  with  a  cast  iron  rod. — A  short  slug  moving  at 
velocity  2  V  comes  squarely  against  the  end  of  a  rod  as  indicated 
by  diagram  A  in  Fig.  91.  Let  us  assume  that  slug  and  rod  are 
of  the  same  diameter  and  made  of  the  same  material.  Then 
as  long  as  the  slug  pushes  on  the  rod  It  may  be  thought  of  as  a 
part  of  the  rod,  and  the  state  of  affairs  as  shown  in  diagram  A 
is  resolvable  into  two  oppositely  moving  waves  (ribbon  waves) 
V  —  C  and  V  +  C  as  shown.  The  wave  V  —  C  merges 
into  V  ~\-  C  as  it  is  reflected  from  the  free  end  of  the  slug,  and 
soon  a  ribbon  wave  of  length  2L  (where  L  is  the  length  of  the 
slug)  develops  and  travels  to  the  right  along  the  rod  as  indicated 
in  diagrams  B  and  C.  This  ribbon  wave  is  reflected  from  the 
right-hand  end  of  the  rod  as  indicated  in  diagram  D,  a  region 
of  doubled  velocity  2V  and  zero  compression  develops  until 
the  ribbon  wave  is  half  reflected  as  shown  in  diagram  E,  and 
then  a  region  R  of  tension  begins  to  develop  as  indicated  in 


THE   DYNAMICS  OF   WAVE   MOTION. 


185 


diagram  F.  This  tension  is  equal  to  the  original  compression, 
and,  if  the  rod  is  made  of  cast  iron  which  withstands  very  great 
compression  but  which  does  not  withstand  great  tension  it  may 
easily  be  that  the  rod  separates  in  the  narrow  region  R  in  dia- 


D 


2V    +C 

1 

c 


Fig.  91. 


j.-.y  *rt7 


-C 


gram  .F.  The  imperfect  elasticity  and  lack  of  homogeneity  of 
a  cast-iron  rod  modifies  the  ideal  action  as  represented  in  Fig.  91, 
and  some  appreciable  time  is  required  for  the  cast-iron  rod  to 
separate  in  the  region  R.  Therefore  the  slug  which  flies  off 
the  end  of  the  cast-iron  rod  is  sure  to  be  somewhat  longer  than 
the  original  slug  in  diagram  A,  and  it  will  be  moving  as  a  whole 
at  a  lower  velocity  than  2  V.  Nevertheless  it  is  a  very  interesting 
and  instructive  experiment  to  shooc  a  steel  slug  against  the  end 
of  a  cast-iron  rod. 

Use  a  No.  10  single-barrel  shot  gun;  use  a  snugly  fitting 
steel  slug  about  2  inches  long;  make  the  cast-iron  rod  of  same 
diameter  as  steel  slug  with  a  squarely  ground  end;  and  use 
about  half-a-gram  of  black  powder  with  an  empty  space  of  two 
inches  or  more  between  powder  and  slug.  This  experiment  is 
suggestively  frightful  but  in  fact  quite  safe. 


186  CALENDAR   OF   LEADING   EXPERIMENTS. 

An  interesting  example  of  the  action  which  is  represented  in 
Fig.  91  is  afforded  by  some  experiments  which 
were  made  at  Woolwich,  England,  to  determine 
the  effect  of  exploding  a  charge  of  dynamite,  or 
other  high  explosive,  against  a  thick  plate  of  steel 
armor.  In  some  of  the  experiments  a  high  bulge 
was  formed  on  the  back  of  the  plate,  and  when 
the  plate  was  cut  open  its  section  was  as  shown 
in  Fig.  92.  A  deep  dent  was  formed  where 

pv.  92  the  explosion  took  place,  a  layer  of  steel  /  was 

thrown  off  the  other  side  of  the  plate,  and  a  hol- 
low space  h  was  left. 

Experiment  with  billiard  balls. — Place  a  number  of  billiard  balls 
together  in  a  straight  row.  Let  one  ball,  two  balls,  three  balls, 
etc.,  come  against  the  end  of  the  row,  and  one  ball,  two  balls, 
three  balls,  etc.,  will  fly  off  from  the  other  end  of  the  row.  The 
action  is  essentially  the  same  as  that  shown  in  Fig.  91. 

Whip  action. — An  interesting  modification  of  the  above  de- 
scribed experiment  of  shooting  a  steel  slug  against  the  end  of  a 
cast-iron  rod  is  to  use  a  tapering  cast-iron  rod  and  shoot  the  steel 
slug  against  the  larger  end  of  the  rod.  As  the  ribbon  wave 
travels  towards  the  smaller  end  of  the  rod  the  values  of  V  and 
C  increase,  and  a  much  lower  initial  velocity  of  slug  is  sufficient 
to  throw  off  the  tip  of  the  tapered  rod. 

This  wave  intensification  along  a  tapered  rod  is  sometimes 
troublesome  in  modern  high-power  guns  which  taper  from  breech 
to  muzzle.  A  quick-acting  backward  force  on  the  breech  block 
produces  a  ribbon  wave  of  tension-and-backward-motion,  and 
as  this  ribbon  wave  travels  towards  the  muzzle  it  is  intensified, 
and  the  tension  may  rise  to  a  value  exceeding  the  strength  of  the 
material,  thus  breaking  off  the  muzzle  of  the  gun,  or,  what  is 
quite  as  bad  for  the  gun,  leaving  a  permanent  stretch  of  the 
muzzle  portions  of  the  gun. 

The  intensification  of  a  wave  which  travels  along  a  tapered 


THE   DYNAMICS  OF   WAVE   MOTION.  187 

rod  is  an  example  of  what  Professor  P.  G.  Tait  called  whip 
action,  and  Professor  Tait's  explanation  of  the  cracking  of  a 
whip  is  as  follows:  First,  any  object  moving  through  the  air 
at  a  velocity  greater  than  the  velocity  of  sound  produces  a 
sharp  snap  or  crack.  Second,  a  wave  of  moderate  sidewise 
velocity  may  be  started  in  the  thick  and  heavy  butt  portions  of 
a  whip,  and  the  sidewise  velocity  increases  more  and  more 
as  the  wave  travels  towards  the  small  end  of  the  whip,  thus  caus- 
ing the  whip  cracker  to  reach  velocities  exceeding  1,000  feet  per 
second. 

An  interesting  experiment  is  to  stand  on  a  ladder,  hold  the 
butt  end  of  a  long  flexible  whip  in  the  hand,  let  the  whip  hang 
vertically  downwards,  move  the  butt  quickly  to  one  side,  and 
watch  the  increasing  visible  amplitude  of  motion  of  the  resultant 
wave  as  it  travels  down  the  whip.* 

Gun  recoil.  Simple  ideal  case. — A  portion  of  the  recoil  of  a 
heavy  gun  (backward  momentum  of  gun,  the  gun  being  assumed 
to  be  free)  is  equal  to  the  forward  momentum  of  the  projectile, 
and  the  remainder  is  equal  to  the  forward  momentum  of  the 
powder  gases.  The  first  part  of  the  recoil  momentum  is  quite 
easily  calculated  whereas  no  rational  method  has  hitherto  been 
developed  for  calculating  the  second  part  of  the  recoil  momen- 
tum, f  A  very  simple  and  more  or  less  ideal  example  of  gas- 
recoil  would  be  afforded  by  suddenly  opening  one  end  of  a  tube 
which  contains  very  slightly  compressed  air.  The  initial  condi- 
tion of  uniform  compression  is  symbolized  by  the  heavy  dotted 
line  in  Figs.  93  and  94,  and  the  heavy  arrow  symbolizes  the  long- 
drawn-out  ribbon  wave  (of  outward-motion-and-raref action, 
+  V  and  —  C)  which  continues  to  shoot  inwards  from  the 
open  end  of  the  tube.  The  first  lap  of  this  ribbon  wave  wipes 
out  the  existing  compression  and  lays  down  a  condition  of  out- 

*  Change  of  wave  velocity  due  to  changing  mass  and  tension  have  an  important 
effect  here,  but  it  is  not  worth  while  to  discuss  the  matter  further. 

f  This  deficiency  has  recently  been  met  in  a  paper  by  Wm.  S.  Franklin,  Journal 
of  the  Franklin  Institute,  Vol.  CLXXIX,  pages  559-577,  May,  1915. 


188 


CALENDAR   OF   LEADING   EXPERIMENTS. 


ward  motion.  The  ribbon  wave  is  reflected  from  the  closed 
end  of  the  tube  with  reversal  of  V,  and  the  second  lap  of  the 
ribbon  wave  wipes  out  the  outward  motion  and  lays  down  a 
condition  of  rarefaction.  The  ribbon  wave  is  then  reflected 


initial  compression  C 
moving 


still 


initial  Compression^  (7 
still 


compressed     [not   compressed  ;' 


-C 


Fig.  93. 


Fig.  94. 


from  the  open  end  with  reversal  of  C,  and  the  third  lap  of  the 
ribbon  wave  wipes  out  the  rarefaction  and  lays  down  a  condition 
of  inward  motion.  The  ribbon  wave  is  then  reflected  from  the 
closed  end  of  the  tube  with  reversal  of  V  and  the  fourth  lap  of 
the  ribbon  wave  wipes  out  the  inward  motion  and  lays  down  a 
condition  of  uniform  compression  as  at  the  beginning. 

If  such  a  simple  wave  motion  existed  in  the  powder  gases  of 
a  gun,  the  gas- recoil  momentum  would  be  due  to  the  fact  that 
the  gases  would  continue  to  push  backwards  on  the  breech  block 
after  the  opening  of  the  muzzle  for  the  length  of  time  required 
for  the  first  lap  of  the  ribbon  wave  to  reach  the  breech  block. 
As  a  matter  of  fact,  however,  the  wave  motion  in  the  powder 
gases  is  greatly  complicated  by  the  velocity  already  existent  in 
the  gases  when  the  muzzle  is  opened  and  by  the  adiabatic  cooling 
of  the  powder  gases  as  they  expand. 

Experiment  with  Kelvin  ladder  showing  the  effect  described 
in  connection  with  Figs.  93  and  94. — Clamp  the  crank  at  the  top 
of  the  ladder,  take  hold  of  the  bar  at  the  bottom  and  twist  the 
ladder  slightly,  and  wait  for  the  ladder  to  become  still.  Then 
release  the  bottom  bar.  The  initial  state  of  twist  is  analogous  to 
the  initial  compression  of  the  air  in  a  tube,  and  the  initial  twist 
may  be  thought  of  as  symbolized  by  the  heavy  dotted  line  in 


THE   DYNAMICS  OF   WAVE   MOTION.  189 

Figs.  93  and  94.  Releasing  the  bottom  bar  is  analogous  to  the 
opening  of  the  end  of  the  tube.  A  ribbon  wave  (of  rotation-and- 
reversed-twist)  then  wipes  out  the  existing  twist  and  lays  down 
a  condition  of  uniform  rotation.  At  the  instant  this  ribbon 
wave  reaches  the  top  of  the  ladder  the  ladder  will  be  seen  to  be 
flat  (entirely  freed  from  twist)  and  the  entire  ladder  will  be  in 
uniform  rotation.  The  further  details  of  behavior  of  ladder 
need  not  be  described  but  they  should  be  fully  described  if  the 
experiment  is  shown  to  a  class,  otherwise  the  student  will  not 
know  what  to  look  for. 

Differential  equations  of  electrical  wave  motion  on  a  trans- 
mission line. — The  theory  of  electrical  wave  motion  is  usually 
developed  in  terms  of  electric  and  magnetic  field  intensities 
in  space,*  and  the  equations  are  not  easy  to  understand,  especially 
by  the  electrical  engineer  who  is  accustomed  to  express  every- 
thing in  terms  of  voltage  and  current.  Therefore  the  following 
discussion  of  electric  wave  motion  on  a  transmission  line  is 
expressed  in  terms  of  voltage  and  current.  Throughout  the 
discussion  the  resistance  of  the  line  wires  is  assumed  to  be 
negligible  and  the  line  wires  are  assumed  to  be  perfectly  insulated. 

The  horizontal  lines  in  Fig.  95  represent  the  wires  of  a  trans- 


wire 


«— >      AX         — >i+Ai  wire 


\a 

b\ 

i 

i 

L 

|e+4e 

wire 

[d 

c; 

wire 

Fig.  95. 

mission  line,  and    abed   is  an  element  of  the  line.     Let   e   be  the 
voltage  across  the  transmission  line  at  the  point    ad    and  let    i 

*  A  very  simple  development  of  this  general  theory  is  given  on  pages  186-195 
of  Franklin's  Electric  Waves,  The  Macmillan  Co.,  1909.  The  student  must  be 
familiar  with  the  elements  of  vector  analysis  to  be  able  to  understand  this  electro- 
magnetic theory.  See  Electric  Waves,  pages  158-185,  or  see  Franklin,  MacNutt 
and  Charles's  Calculus,  pages  210-253. 


190  CALENDAR  OF  LEADING   EXPERIMENTS. 

be  ''the  current  in  the  line  at  the  same  point"  (meaning  outflow- 
ing current  in  one  wire  and  returning  current  in  the  other  wire) 
as  shown  in  Fig.  95.  The  voltage  across  the  line  at  the  point  be 
is  e  +  Ae,  and  the  current  in  the  line  at  the  same  point  is 
i  +  At. 

The  capacity  of  the  element  abed  is  C-Ax,  where  C  is  the 
capacity  of  unit  length  of  the  line.  Therefore  the  charge  q 
"on  the  element"  (positive  charge  on  ab  and  negative  charge  on 
cd)  is  q  =  C-Ax  X  e,*  and  the  rate  of  decrease  of  g,  namely, 

—  —  "C'Ax,    is  equal  to   Ai.     Therefore  we  have: 

•cde-      -  in 

cdi~    ~  ex 

The  net  electromotive  force  around  the  elementary  circuit 
abed  is  (e  +  Ae)  —  e,  and  this  electromotive  force  causes  the 
current  in  the  circuit  to  decreasef  at  a  definite  rate  such  that 

di 

Ae  =  —L'&xX—,      according  to  Art.    196,   pages  276-277, 
ot 

General  Physics,  where  L  is  the  inductance  per  unit  length  of  the 
transmission  line  and  L-kx  is  the  inductance  of  the  elementary 
circuit  abed.  Therefore  we  have 

di  de 


Equations  (i)  and  (ii)  contain  the  two  unknown  dependent 
variables  e  and  i,  and  it  is  necessary  to  eliminate  one  to  get 
an  equation  involving  the  other  alone.  By  differentiating  equa- 
tion (i)  with  respect  to  /  and  equation  (ii)  with  respect  to  x 

*  The  charge  is  greater  than  C-Ax  X  e  and  less  than  C-Ax  X  (e  +  Ae),  and 
when  Ax  approaches  zero,  the  expression  for  q  approaches  C-Ax  X  e  as  a  limit. 

t  The  electromotive  force  e  +  Ae  is  associated  with  an  electric  field  from  wire 
to  wire  and  the  arrow  shows  the  direction  of  this  field  or  the  direction  of  e  +  A« 
as  it  would  be  indicated  by  a  voltmeter.  Evidently,  however,  an  excessive  charge 
on  the  wires  at  be  and  a  large  electromotive  force  from  b  to  c  would  tend  to 
create  a  current  opposite  to  i. 


THE   DYNAMICS  OF  WAVE  MOTION.  191 

we  get 

r—  -          d2*  dH          _d% 

CW  =     ~  dx-dt    '  dt-dx  ~     "  dx* 

but   d2i/(dx-dt)  =  d2i/(dt-dx)*   and  therefore  we  get 

d*e        i      d*e 
dfi  "LC'dx2 

By  differentiating  equation  (it)  with  respect  to  /  and  equation 
(i)  with  respect  to   x,   and  eliminating  as  before,  we  get 

^  _  1?_   ¥i 
dt*  ~  LC'dx* 

The  general  solution  of  equation  (10)  is  of  course 

e  =  F(x  -  vt)  +  f(x  +  vt) 
where 


but  it  is  most  convenient  for  present  purposes  to  consider  the 
two  particular  solutions  e  =  F(x  —  vt)  and  e  =  f(x  +  vt) 
as  follows: 

A  pure  wave  traveling  to  the  right  in  Fig.  95  is  represented  by 
the  particular  solution 

e  =  F(x  -  vt)  (13) 

and  to  adopt  this  solution  of  (10)  is  to  fix  the  corresponding 
solution  of  (u)  as  follows:  From  equations  (i)  and  (13)  we  have 

-  Cv-F'(x  -  vt)  =  -  j-  (iii) 

and  from  equations  (ii)  and  (13)  we  have 

L  -^  =  -  F'(x  -  vt)  (iv) 

*  This  is  by  no  means  self-evident  as  an  arithmetical  proposition;    but  a  full 
discussion  of  it  would  be  out  of  place  here. 


192  CALENDAR  OF   LEADING   EXPERIMENTS. 

Therefore,  by  integrating  (iii)  and  (iv),  we  get 

i  =  Cv  •  F(x  —  vt)  +  any  function  of  t  (v) 

and 

i  =  j-  •  F(x  —  vt)  +  any  function  of   x  (vi) 

Lit) 

i  fC 

But,  according  to  equation  (12),  Cv  =  -=-  =  +  yj- ,  and  there- 
fore the  unknown  function  of  /  in  (v)  must  be  identical  to  the 
unknown  function  of  x  in  (vi),  which  means,  in  the  most 
general  case,  that  both  functions  are  constants;  and  for  present 
purposes  these  constants  may  be  taken  as  zero.  Therefore 
equations  (v)  and  (vi)  become 

i  =  a-F(x-vt)  (14) 

where 


'L 
whence  dividing  (14)  by  (13)  we  get 

;  -  +  a  (16) 

e 

From  equations  (13),  (14)  and  (15)  it  is  evident  that  a  pure 
wave  traveling  to  the  right  in  Fig.  95  is  any  distribution  of  voltage 
e  traveling  to  the  right  at  velocity  v,  the  current  at  each  point  in  the 
line  being  equal  to  ae;  or  any  distribution  of  current  i  traveling 
to  the  right  at  velocity  v,  the  voltage  across  the  line  at  each 
point  being  equal  to  i/a. 

A  pure  wave  traveling  to  the  left  in  Fig.  95. — Starting  with 
e  =  f(x  +  vt)  we  may  find,  by  an  argument  similar  to  the  above, 
that  i  —  —  ae.  That  is 

\  =  -  a  (17) 

and  it  follows  that  a  pure  wave  traveling  to  the  left  in  Fig.  95  is 
any  distribution  of  voltage  e  traveling  to  the  left  at  velocity  v,  the 
current  at  each  point  in  the  line  being  equal  to  —  ae. 


THE   DYNAMICS  OF   WAVE   MOTION.  193 

Remark. — Equations  (16)  and  (17)  are  useful  in  that  they 
enable  one  to  pick  out  particular  solutions  of  equations  (io)and 

en). 

Above  discussion  from  another  point  of  view. — Professor 
P.  G.  Tait's  classical  discussion  of  wave  motion  on  a  stretched 
string  which  does  away  with  all  necessity  for  the  solution  of 
differential  equations*  has  its  analog  in  the  theory  of  electrical 
waves,  and  it  is  interesting  to  note  that  the  avoidance  of  integra- 
tion comes  from  the  postulate  of  travel  which  is  introduced  at 
the  start.  This  fact  makes  it  all  the  more  clear  that  the  classical 
differential  equation  of  wave  motion  [see  equation  (10)  or  (n) 
above]  is  merely  a  differential  equation  of  travel. 

Effect  of  traveling  distribution  of  current. — Imagine  current  to 
be  distributed  over  a  transmission  line  in  any  arbitrary  manner, 
the  current  i  at  any  given  point  ad  of  the  line  (meaning  out- 
flowing current  i  in  one  wire  and  back  flowing  current  i  in 
the  other  wire)  being  represented  by  the  prdinate  i  of  any 
given  curve  CC,  Fig.  96,  and  suppose  this  current  distribution 
to  travel  as  a  whole  to  the  right  in  Fig.  96  at  velocity  v,  that  is 
to  say,  let  us  imagine  the  curve  CC  to  travel  to  the  right  at 
velocity  v  and  the  current  distribution  to  change  so  as  to  be 
represented  at  each  instant  by  this  traveling  curve.  It  is  strictly 
meaningless  to  think  of  the  current  itself  as  moving  along,  but 
it  is  convenient  to  think  of  the  current  and  the  magnetic  field 
(also  the  charges  on  the  wires  and  the  electric  field  between  the 
wires)  as  moving  along  with  CC.  Such  a  traveling  current  dis- 
tribution would  produce  a  voltage  distribution  (traveling  along 
with  it)  such  that 

e  =  Liv  (18) 

where  e  is  the  voltage  across  the  line  at  the  place  where  the 
current  in  the  line  is  i.  This  equation  may  be  established  as 
follows :  The  inductance  of  the  element  abed  is  L  •  Ax,  and  the 
magnetic  flux  between  the  wires  ad  and  be  is  equal  to  the 

*  See  General  Physics,  Art.  359,  pages  499-501. 
14 


194 


CALENDAR   OF   LEADING   EXPERIMENTS. 


inductance  of  the  element  multiplied  by  the  current,  everything 
being  expressed  in  c.g.s.  units.  Now,  the  current  distribution 
and  the  associated  flux  are  assumed  to  travel  to  the  right  at 
velocity  v  so  that  all  of  the  flux  between  ad  and  be  will  sweep 
across  the  line  be  in  Ax/v  of  a  second.  Therefore  the  voltage 
e  induced  along  be  by  the  traveling  flux  is  Li- Ax  divided  by 
Ax/v  which  gives  Liv  abvolts. 

Effect   of  traveling   distribution   of  voltage. — Imagine   electric 
charge  to  be  distributed  over  the  transmission  line  in  Fig.  96 


wire 


wire 


Fig.  96. 

(positive  charge  on  one  wire  and  an  equal  negative  charge  on 
the  other).  This  charge  means  a  definite  voltage  between  wires 
at  each  point  along  the  line,  and  we  may  imagine  the  voltage  e 
at  each  point  to  be  represented  by  the  ordinate  of  a  curve  CC. 
Imagine  the  electric  charge  and  the  associated  voltage  distribu- 
tion to  travel  along  the  line  at  velocity  v.  Such  a  traveling 
voltage  distribution  would  produce  a  definite  current  distribution 
over  the  line  such  that 

i  =  Cev  (19) 

where  i  is  the  current  in  the  line  at  the  place  where  the  voltage 
across  the  line  is  e,  and  C  is  the  capacity  of  the  line  per  unit 
of  length.  This  equation  may  be  established  as  follows:  The 
capacity  of  the  element  ab  cd  is  C-Ax  so  that  C-Ax  X  e  is 
the  positive  charge  on  ab  (or  negative  charge  on  cd).  But 
all  of  the  charge  on  ab  must  flow  past  the  point  b  during 


THE   DYNAMICS  OF   WAVE   MOTION. 


195 


Ax/v  of  a  second,  so  that  the  current  in  the  line  is  found  by 
dividing  C-kx  X  e  by  Ax/v  which  gives  equation  (19).  Of 
course  the  current  distribution  travels  along  with  voltage  dis- 
tribution which  produces  it. 

Mutually  sustaining  current  and  voltage  distributions. — Let  e,  i 
and  v  refer  to  identically  the  same  quantities  in  equations  (18) 
and  (19)  respectively,  that  is  to  say,  e  is  produced  by  the  motion 
of  i,  and  in  turn  i  is  produced  by  the  motion  of  e  (see  pages 
100-105  of  this  volume).  Then  equations  (18)  and  (19)  are 
simultaneous  equations,  and  by  elimination  we  get 


i 
LC 


and 


(20) 
(21) 


A  clear  understanding  of  an  electric  wave  traveling  along  a 
transmission  line  may  be  obtained  from  Fig.  97,  resistance  of 


wire       •»      •*    — *.  —*• 


current 


wire 


x 

^ 

" 

. 

- 

:  HKffl 

:[{::  ::::;:  - 

; 

• 

1 

' 

I 

+  charge' 

X 

« 

• 

• 

•   '!!;;'•  ill 

;:••:::;;  i   . 

* 

• 

, 

.' 

V   ^ 

—  charge^ 

^  •  ' 

. 

. 

/ 

• 

•     ;'.  •  :  ::  :  •:': 

iiji-i-:'   •' 

; 

^ 

• 

'• 

wire 

S 

v 

•+— 

•  ^ 

1— 

•^ 

— 

•:ji:.;;j  •  •  •- 

h- 

/ 

« 

* 

wire 

current 


Fig.  97. 
Showing  an  electromagnetic  wave  on  a  transmission  line. 

line  wires  being  negligible  and  wires  being  perfectly  insulated. 
The  curve  WW  corresponds  to  CC  in  Fig.  96,  and  the  ordi- 
nate  y  represents  the  voltage  e  across  the  line  or  the  current  i 


196 


CALENDAR  OF   LEADING  EXPERIMENTS. 


in  the  line  at  the  point  p.  The  upper  wire  is  positively  charged 
and  the  fine  vertical  lines  represent  the  lines  of  force  of  the  electric 
field  which  emanates  from  the  positively  charged  wire  and  con- 
verges upon  the  lower  wire  which  is  negatively  charged.  Cur- 
rent flows  to  the  right  in  the  upper  wire  and  to  the  left  in  the 
lower  wire  as  represented  by  the  short  horizontal  arrows,  and 
the  fine  dots  represent  the  lines  of  force  of  the  magnetic  field 
between  the  wires,  this  magnetic  field  being  perpendicular  to  the 
plane  of  the  paper  in  Fig.  97.  The  heavy  arrow  shows  the 
direction  of  travel  of  the  wave  at  velocity  v. 

The  direction  of  travel  and  the  directions  of  e  and  i  may  be 
correlated  as  follows:  That  particular  wire  is  positively  charged 
out  of  which  the  electric  lines  of  force  emanate;  the  voltage  e  is 
from  positively  charged  wire  to  negatively  charged  wire;  and 
the  current  in  the  positively  charged  wire  may  be  thought  of  as 
carrying  the  positive  charge  forwards  from  the  back  of  the  wave 
and  laying  it  down  on  the  wire  in  front  of  the  wave. 

What  determines  wave  form? — A  device  which  produces  a 
varying  voltage  e  and  which  can  deliver  current  ad  libitum  is 
connected  across  the  end  of  a  transmission  line,  and  the  curve  in 
Fig.  99  (which  is  of  course  traveling  to  the  right)  which  represents 
the  shape  of  the  wave  which  shoots  out  along  the  transmission 
line  is  exactly  like  the  curve  in  Fig.  98  which  shows  e  as  a 


axis  of  e 


axis  of  time 


axis  of  e 


axis  of  x 


wire 


transmission  line 


Fig.  98. 

Showing  voltage    e    as  a  function 
of  the  time. 


wire 

Fig.  99. 

Showing  shape  of  wave  which  shoots  out 
from  end  of  a  transmission  line  when  volt- 
age e  is  connected  across  end  of  line.  The 
wave  involves  a  certain  current  /  =  ae  at 
each  part  of  line  according  to  equation  (16). 


THE   DYNAMICS  OF   WAVE   MOTION. 


197 


function  of  the  time.  This  is  evident  when  we  consider,  first, 
that  the  assumed  wave  satisfies  the  differential  equations  (10) 
and  (n),  second,  that  the  assumed  wave  is  consistent  with 
the  fact  that  the  transmission  line  is  entirely  quiescent  up  to 
the  instant  of  connecting  e,  and  third,  that  the  assumed  wave 
in  passing  out  on  the  line  in  Fig.  99  involves  at  each  instant  a 
voltage  e  across  the  end  of  the  line  which  is  in  fact  equal  to 
the  voltage  acting  at  that  instant  according  to  Fig.  98.  That 
is  to  say,  first,  the  differential  equations  are  satisfied,  second,  the 
initial  conditions  are  satisfied,  and  third,  the  boundary  conditions 
are  satisfied. 

The  wave  train. — When  a  periodic  electromotive  force  acts 
across  the  end  of  the  line,  for  example  when  an  alternator  is 
connected  to  the  line,  then  what  is  called  a  wave-train  passes 
out  along  the  line,  and  the  state  of  affairs  (before  matters  are  com- 
plicated by  the  reflection  of  the  waves  from  the  distant  end  of 
the  line)  is  shown  in  Fig.  100.  This  figure  shows  a  wave- train 


direction  of 


progression 


Fig.  100. 

which  is  produced  by  a  harmonic  alternating  voltage,  and  the 
curve  WW  is  a  curve  of  sines.  The  short  horizontal  arrows 
represent  the  current  at  various  places  in  the  wires,  the  fine 
vertical  lines  represent  the  lines  of  force  of  the  electric  field, 
and  the  dots  represent  the  lines  of  force  of  the  magnetic  field 
as  in  Fig.  97. 

The  ribbon  wave. — When  a  battery  of  constant  voltage  and 
negligible  resistance  is  connected  across  the  end  of  a  transmission 


198 


CALENDAR   OF   LEADING   EXPERIMENTS. 


line,  a  wave  shoots  out  on  the  line,  the  voltage  e  in  the  wave  is 
everywhere  of  the  same  value  and  equal  to  battery  voltage,  and 
the  current  is  everywhere  the  same  in  value  and  equal  to 
e  X  ilC/L,  according  to  equation  (16).  Such  a  wave  we  will 
call  a  ribbon  wave.  Thus  Fig.  101  represents  a  ribbon  wave  d/v 


wire 


ribbon   wave 

Fig.  101. 

Showing  the  ribbon  wave  which  shoots  out  from  a  battery  which  is  suddenly 
connected  to  the  end  of  a  line. 

seconds  after  the  battery  is  connected.  The  short  horizontal 
arrows  represent  current,  the  fine  vertical  lines  represent  the 
lines  of  force  of  the  electric  field,  and  dots  represent  magnetic 
lines  of  force  as  in  Figs.  97  and  100. 

Reflection. — For  the  sake  of  brevity  and  clearness  we  will 
discuss  only  the  reflection  of  the  ribbon  wave,  and  such  a  wave 
will  be  represented  by  a  single  heavy  horizontal  arrow  as  in  Fig. 
101.  When  such  a  wave  is  turned  back  or  reflected  at  the  end 
of  a  line,  the  heavy  arrow  will  be  shown  as  turned  back  as  in 
Figs.  102,  103  and  104.  The  voltage  and  current  in  the  original 
wave  are  represented  by  E  and  /  and  the  voltage  and  current 
in  the  reflected  wave  are  represented  by  Er  and  Ir  as  shown. 

Reflection  from  open  end  of  line. — The  doubled  arrow  in  Fig. 
1 02  represents  a  wave  which  has  been  turned  back  or  reflected 
at  the  open  end  of  a  transmission  line.  The  necessary  condition 
which  must  be  satisfied  at  the  open  end  of  a  line  is  that  the  actual 
current  there  be  zero.  Therefore  we  have: 


/  +  /,  =  o 


THE   DYNAMICS  OF   WAVE   MOTION.  199 

In  addition  to  this  we  must  have : 

E  JL 

7  =  +  \  r  M 

J.  *  Lx 

and 

f=->i  (iii) 

Now  the  voltage  and  current  in  the  original  wave  are  supposed 
to  be  given  and  the  voltage  and  current  in  the  original  wave  do, 
as  a  matter  of  course,  satisfy  equation  (ii).  From  equation  (i) 
we  have: 

Ir  =  -  /  (iv) 

and  from  equations  (ii)  and  (iii)  we  have: 

Er  =  +  E  (v) 

Reflection  at  the  open  end  of  a  line  is  therefore  complete  and  it 
takes  place  with  reversal  of  current. 

wire 


open   end 
wire                          of  H™ 

wire 

short-circuited 
end  of  line 

--m    E       / 

.     v    V     X       E                        1 

Fig.  103. 

Showing  a  ribbon  wave  partly  reflected      Showing  a  ribbon  wave  partly  reflected 
from  the  open  end  of  a  line.  from  the  short-circuited  end  of  a  line. 

Reflection  from  short-circuited  end  of  line. — The  doubled  arrow 
in  Fig.  103  represents  a  wave  which  has  been  turned  back  or 
reflected  from  the  short-circuited  end  of  a  line.  The  necessary 
condition  which  must  be  satisfied  at  the  short-circuited  end  of  a 
line  is  that  the  actual  voltage  across  the  end  be  zero.  Therefore 

we  have: 

E  +  Er  =  o  (vi) 

whence 

Er  =  -  E  (vii) 


2OO 


CALENDAR  OF  LEADING  EXPERIMENTS. 


and,  since  equations  (ii)  and  (iii)  always  apply,  we  get: 

Ir   =    +  /  (Viii) 

Reflection   at  a   short-circuited   end   of  a   transmission   line   is 
therefore  complete  and  it  takes  place  with  reversal  of  voltage. 

Transmission  line  oscillations  which  follow  the  switching  on 
of  a  generator. — When  a  generator  of  negligible  resistance  and 

wire 


wire 


open  end  of  line 


+E 


-E        +1 


+E       +1 


axis  of  I 


axis  of  time 


curve  of  current  at  B 

Fig.  104. 

The  upper  part  of  the  figure  shows  a  ribbon  wave  which  shoots  out  from  a 
suddenly  connected  generator  and  is  repeatedly  reflected  at  both  ends  of  the  line. 
The  lower  part  of  the  figure  shows  the  value  of  the  current  at  B  as  a  function  of 
elapsed  time. 

inductance  is  suddenly  switched  on  to  a  line,  a  ribbon  wave  of 
generator  voltage  and  corresponding  current  shoots  out  over  the 
line.  .  Assuming  line  resistance  to  be  negligible  and  line  insulation 
to  be  perfect,  this  ribbon  wave  is  reflected  back  and  forth  as 
represented  in  Figs.  104  and  105,  and  by  adding  voltages  and 


THE   DYNAMICS  OF   WAVE   MOTION. 


201 


currents  in  the  successive  laps  of  the  ribbon  wave  a  precise 
knowledge  of  the  distribution  of  voltage  and  current  over  the 
line  at  any  instant  may  be  obtained.  Thus,  after  a  thousandth 
of  a  second,  the  total  length  of  the  ribbon  wave  would  be  186 

short-circuited  end   of  line 


wire 


wire 


x 


=1                       wire 

wire 

s\  \\vvx     ribbon   wave                    i-E             +1 

-E             +1 

) 

L                                                         i-E            +1 

-E            +1 

) 

axis  of  I 


curve  of    growth   of 
current  at  B 


axis  of   time 


Fig.  105. 
Same  as  Fig.  104  except  that  distant  end  of  line  is  short-circuited. 

miles  which  would  give  a  definite  number  of  complete  laps  and 
a  fraction  of  the  next  lap.  By  adding  a  number  of  laps  in  this 
way,  curves  can  be  plotted  showing  the  actual  distribution  of 
voltage  and  current  over  the  line  at  any  given  instant;  or  values 
of  current  and  voltage  at  a  given  point  of  the  line  can  be  found  for 
successive  instants  and  from  this  data  curves  can  be  plotted 
showing  voltage  or  current  at  any  point  as  functions  of  elapsed 


2O2 


CALENDAR   OF   LEADING   EXPERIMENTS. 


times.     The  ampere-time  curves  in  the  lower  parts  of  Figs.  104 
and  105  were  obtained  in  this  wav 


wire 


wire 


wire 


wire 


R 


•M 


-kE 


-il 


axis  of  I 


axis  of  I 


curve 'of  current  growth   at  B       curve  tf  current  growth  at  R 

Fig.  106. 

Same  as  Fig.  104  except  that  distant  end  of  line  is  connected  to  a  non-inductive 
circuit  of  which  the  resistance  is  R  =  3  v—  .  In  this  case  the  ribbon  wave  is  only 

partially  reflected  from  the  end   R   and  the  successive  laps  of  the  ribbon  wave  are 
thereby  greatly  reduced  in  intensity  as  represented  by  the  shading. 

Note. — The  character  of  the  reflection  at  B  in  Figs.  104,  105 
and  106  is  determined  by  the  condition  that  the  sum  of  the  volt- 
ages in  the  successive  laps  must  always  be  equal  to  battery 


THE   DYNAMICS  OF   WAVE   MOTION.  203 

voltage     E,     and  of  course  there  is  always  an  odd  number  of 
laps  at   B,    one,  or  three,  or  five,  etc. 

Note. — Figures  104,  105  and  106  show  what  takes  place  when 
a  direct-current  generator  or  battery  is  switched  on  to  the  line, 
but  on  a  line  of  moderate  length  a  number  of  laps  are  formed  in  an 
excessively  short  interval  of  time,  and  therefore  Figs.  104,  105  and 
1 06  show  quite  accurately  what  takes  place  immediately  after  an 
alternator  is  switched  on  to  the  line,  E  being  the  value  of  the  voltage 
of  the  alternator  at  the  instant  of  closing  the  switch. 

Transmission  line  surges  which  are  produced  when  a  circuit 
breaker  opens. — When  a  circuit  breaker  opens,  the  arc  which 
is  formed  persists  for  a  very  long  time,  relative  speaking,  and 
the  open  gap  in  the  circuit  is  filled  with  a  fairly  good  conducting 
material  which  slowly  loses  its  conductivity.  It  is  about  as 
nearly  impossible  to  produce  characteristic  line  surges  by  opening 
a  circuit  breaker  as  it  would  be  to  set  up  an  abrupt  water  wave 
in  a  canal  by  allowing  a  cubic  mile  of  soft  mud  to  flow  into  the 
canal  prism  to  stop  a  troublesome  flow  of  water  in  the  canal ;  and 
yet  the  moon,  as  a  6o-cycle  generator  (60  cycles  per  month!), 
might  produce  a  troublesome  tidal  wash  in  a  large  estuary  while 
the  attempt  was  being  made  to  "open-circuit"  the  estuary  in 
this  Brobdignagian  fashion!  Let  the  reader  consider  this 
hydraulic  analog  carefully.  It  reproduces  nearly  all  of  the  es- 
sentials of  the  electrical  case.  The  conducting  vapor  in  the  arc 
of  a  circuit  breaker  is  somewhat  analogous  to  mud  as  a  dam- 
building  material. 

Very  little  need  be  said  of  characteristic  line  surges  in  connec- 
tion with  opening  of  switches,  except  in  the  case  of  very  long  lines. 
When  a  line  is  short,  or  only  moderately  long,  what  takes  place 
may  be  described  quite  accurately  in  terms  of  the  simple  ideas 
of  the  elementary  alternating-current  theory,  where  current 
values  are  supposed  to  rise  and  fall  simultaneously  throughout 
an  entire  circuit.  The  formation  of  a  long  arc  between  line  wires 
in  air  and  the  quick  snapping  out  of  such  an  arc  does,  however, 


204  CALENDAR  OF   LEADING   EXPERIMENTS. 

produce  an  electric  wave  disturbance  on  a  transmission  line  of 
moderate  length,  and  the  essential  features  of  this  case  are  shown 
in  Fig.  107.  Imagine  the  system  to  be  short-circuited  by  an  arc 

/*~large  inductance 
S  wire  wire 

V     000000 

©  D 

T    OQOOOQ    wire  _____         wire 


fjg  -I 

+E  +1 


+1 


Fig.  107. 

The  dotted  line  AB  symbolizes  the  initial  current  in  the  transmission  line. 
The  long  heavy  arrow  with  laps  represents  the  ribbon  wave  which  comes  from  D 
after  the  line  at  D  is  opened. 

at  the  distant  end  D  of  the  transmission  line,  voltage  being 
reduced  to  a  negligibly  small  value  over  the  whole  line,  and  a 
large  current  I  being  established  in  the  line.  The  generator  is 
to  be  thought  of  as  having  a  large  inductance  so  that  the  generator 
current  cannot  change  perceptibly  during  the  very  short  time 
required  for  the  characteristic  line  surges.  When  the  distant 
end  of  the  line  is  opened,  a  ribbon  wave  shoots  towards  the  gen- 
erator, is  completely  reflected  at  the  generator  with  reversal  of 
current,  again  completely  reflected  at  the  opened  end  D  of  the 
line  with  reversal  of  current,  and  so  on.  The  first  lap  of  this 
ribbon  wave  wipes  out  the  current  in  the  line  and  lays  down  a 
certain  voltage  E  ( =  A/L/C  times  the  initial  value  of  the 
current) .  The  second  lap  of  the  ribbon  wave  lays  down  a  double 
voltage,  and  the  original  current.  The  third  lap  wipes  out  the 
current  again,  and  lays  down  a  voltage  equal  to  $E,  and  so  on. 

Note. — The  character  of  the  reflection  at  the  generator  in  Fig. 


THE  DYNAMICS  OF  WAVE  MOTION.  205 

107  is  determined  by  the  condition  that  che  sum  of  the  currents 
in  the  successive  laps  at  A  must  always  be  equal  to  the  initial 
current  /  because  the  current  in  the  highly  inductive  generator 
does  not  have  time  to  change  perceptibly. 

Note. — It  must  not  be  imagined  that  the  sending  out  of  a 
ribbon  wave  depends  upon  a  continued  supply  of  energy  at  the 
point  where  the  ribbon  wave  originates.  Thus  in  Fig.  107  the 
ribbon  wave  is  superposed  upon  the  initial  current  /,  and  the 
first  lap  of  the  ribbon  wave  wipes  out  this  current.  Therefore 
since  the  current  is  zero,  there  is  no  energy  flow  at  all  from  the 
opened  end.  The  second  lap  of  the  ribbon  wave  lays  down  the 
original  current  /  and  a  doubled  voltage  2E,  and  this  combina- 
tion of  voltage  and  current  represents  a  flow  of  energy  from  the 
generator  into  the  line. 


THE  TRADITIVE  LAMP. 

Long  ago  Bacon  gave  a  list  of  the  things  needed  for  the  Ad- 
vancement of  Learning,  and,  among  other  things,  he  mentioned 
A  Traditive  Lamp,  or  the  Proper  Method  of  Handing  Down  the 
Sciences  to  Posterity.  This  Lamp  has  not  yet  been  discovered ! 


206 


APPENDIX. 
A  VISITORS'  LABORATORY  OF  PHYSICS. 

The  authors'  experience  shows  that  the  average  visitor  who 
comes  into  the  physical  laboratory  of  a  college  or  technical  school 
is  greatly  interested  in  the  simplest  kind  of  a  non-dazzling  experi- 
ment if  the  experiment  has  a  rational  appeal  and  if  this  rational 
appeal  is  set  forth  in  a  simple  explanation.  Indeed  the  interest 
which  the  average  visitor  takes  in  genuine  natural  philosophy 
has  led  the  authors  to  devote  a  great  deal  of  time  to  visitors. 
A  number  of  simple  experiments,  varied  from  time  to  time,  are 
always  set  up,  and  every  member  of  the  department,  including 
Clarence  and  Pete,  has  taken  a  share  of  the  fun  of  edifying 
visitors. 

This  phase  of  the  museum  idea  is  certainly  important,  and  the 
authors  believe  that  our  generously  supported  public  institutions, 
especially,  should  have  fully  equipped  Visitors1  Laboratories  of 
Physics  and  provide  for  the  work  of  demonstration  by  members 
of  the  Physics  Department  staff. 

The  authors  have  tried  all  of  the  following  experiments  on 
visitors,  some  of  them  many  times,  and  the  greater  portion  of 
these  experiments  could  be  set  up  permanently  in  a  space 
25  feet  X  80  feet  with  two  or  three  small  dark  rooms  adjoining, 
and  the  entire  equipment  need  cost  no  more  than  four  or  five 
thousand  dollars. 

VISITORS'   EXPERIMENTS. 

5.  Coin  and  card  experiment. 

17  and  1 8.     The  bicycle  wheel  as  a  gyroscope. 

19.  Curious  gyroscope  toy. 

20.  Sharp  and  blunt  pointed  tops  and  hard-boiled  egg. 
22.  Model  of  Schlick  balancer.     Gyroscope  oscillations. 

207 


208  CALENDAR  OF   LEADING   EXPERIMENTS. 

24.  Experiment  with  pivoted  stool. 

33.  Gum  camphor  on  water. 

34.  Experiments  relating  to  oil  flotation. 

35.  The  sensitive  flame. 

36.  The  spit-ball  experiment. 

37.  The  water  hammer. 

38.  The  disk  paradox  and  its  water  analog. 

39.  The  jet  pump  and  the  atomizer. 

40.  Ball  riding  on  air  or  steam  jet. 

41.  Experiment  on  ship  suction. 

43.  Curved  flight  of  spinning  ball. 

44.  High  foul. 

46.  The  Brownian  motion. 
50.  The  Trevelyan  rocker. 

53.  Prince  Rupert's  drops. 

54.  The  fire  syringe. 

55.  Cloud  formation  experiment  with  large  flask. 

58.  Crystallization  of  NH4C1  and  of  KC1O3. 

59.  The  boiling  paradox. 
63.  Wood's  metal  spoon. 

66.  The  recalescence  of  steel. 

67.  Retarded  transformations.     Hardening  and  tempering  of 

steel.     Exhibit  sample  of  partly  crystallized  candy. 

70.  Thermo-elastic  properties  of  rubber. 

71.  Large  electro-magnet;   the  magnet  should  have  pole  pieces 

coming  near  together,  (a)  Show  powerful  attraction  for 
a  piece  of  iron ;  (b)  Show  eddy-current  damping  of  motion 
of  thick  sheet  of  copper ;  (c)  Show  eddy-current  damping 
of  motion  of  closed  copper  ring;  and  (d)  Show  absence 
of  eddy-current  damping  of  motion  of  open-circuited 
copper  ring. 

84.  Lead  tree  experiment. 

88.  Conductivity  of  glass  at  high  temperatures. 

91.  Transformer  experiment. 
100.  Electric  dancers. 


APPENDIX.  209 

105.  Electric  oscillations  maintained  by  an  electric  arc. 
108.  Rosin  experiment. 
113.  The  electric  doubler. 

115.  The  corona  discharge. 

116.  Smoke  deposition  by  Cottrell's  method. 

117.  The  ozonizer. 

Discharge  of  electricity  through  gases,  (a)  Geissler  tube 
discharge,  (b)  Crookes  tube  and  cathode-ray  shadow, 
(c)  Magnetic  deflection  of  cathode  rays,  (d)  Heating 
effect  of  cathode  rays,  (e)  Luminescence  by  cathode  rays, 
(/)  X-rays  and  fluoroscope,  (g)  Ionizing  action  of  X- 
rays,  (h)  Ionizing  action  of  alpha,  beta  and  gamma  rays, 
(i)  The  spinthariscope,  and  (j)  Fog  chamber  for  showing 
paths  of  alpha  particles.  See  pages  139  and  140  of  this 
volume. 

1 1 8.  Unit  areas  of  sense  of  touch. 

121.  Looking  at  bright  point  under  water. 

Phantom  boquet.  A  large  concave  mirror  projects  on  top 
of  an  empty  base  a  real  image  of  a  hidden  boquet. 

122.  Visible  beam  beyond  a  large  short-focus  single  lens. 

123.  Model  of  compound  microscope. 

124.  Model  of  simple  telescope. 

125.  Vision.     Looking  through  a  pin  hole  at  a  cord. 

126.  Vision.     The  coin  box. 

127.  Vision.     Seeing  an  object  inverted  when  its  image  is  erect. 

128.  Vision.     Curious  effect  of  two-eye  vision. 

129.  Vision.     Shadows  of  blood  vessels  on  retina. 

130.  Vision.     The  stroboscope. 

131.  Vision.     Reversal  of  sense  of  motion. 

134.  Astigmatism  of  simple  lens  of  narrow  aperture. 

Looking  at  distorted  drawings  in  cylindrical  and  conical 
mirrors.  See  catalogue  of  Max  Kohl. 

136.  Chromatic  aberration  of  the  eye. 

137.  Projection  of  large  image  of  sun  by  telescope   used  as  a 

telephoto-lens. 


210  CALENDAR  OF  LEADING  EXPERIMENTS. 

138.  Spectroscope  demonstrations,  (a)  Continuous  spectrum, 
(b)  Bright-line  spectrum,  (c)  Reversal  of  sodium  lines, 
and  (d)  Dark-line  spectrum  (solar  spectrum). 

140.  Spectroscopic  analysis  of  light  reflected   from  thin   film. 

(a)  Very  thin  soap  film,  (6)  Moderately  thick  mica  plate. 

141.  Looking  at  flaming  arc  through  coarse  diffraction  grating. 

143.  Experiments  on  polarized  light,     a,  b,  c,  d,  e,  f  and  g. 

144.  Experiments  on  color,     a,  6,  c,  d  and  e. 

145.  Reversal  of  order  of  sound  sensations. 

146.  Musical  sticks. 

147.  The  Galton  whistle.     Determination  of  upper  pitch  limit 

of  audibility. 

148.  Simple  modes  of  vibration  of  string. 

149.  Simple  modes  of  vibration  of  air  column.     Long  glass  tube 

whistle  with  lycopodium  powder. 

150.  Chladin's  figures. 

151.  Resonance,     (a)  With  piano  and  voice,   (b)  With  tuning 

fork  and  adjustable  air  column.  A  most  striking  experi- 
ment is  to  connect  a  telephone  to  an  alternating  current 
supply  (the  frequency  must  be  about  150  cycles  per 
second  to  give  best  results)  and  hold  the  telephone  over 
the  mouth  of  a  tube  which  is  slowly  lowered  in  a  jar  of 
water. 

152.  Experiments  on  vowel  sounds. 

153.  The  action  of  the  ear  in  the  perception  of  tone  color  or 

timbre. 

154.  Beats. 

155.  Combination  tone. 

Note. — Details  of  experiments  are  described  in  main  body  of  this  volume  (under 
given  number)  and  in  Franklin  and  MacNutt's  General  Physics. 


FRANKLIN,  MAcNurr  AND  CHARLES 

Publishers  of  Educational  Books 

That  our  interest  in  selling  books  is  not  narrowly  commercial 
is  evident  from  the  wide  departures  from  traditional  points  of 
view  in  every  one  of  the  twenty-six  volumes.  This  number  in- 
cludes Nichols  and  Franklin's  Elements  of  Physics,  three  volumes, 
and  Franklin  and  Williamson's  Elements  of  Alternating  Currents. 
We  are  propagandists  rather  than  commercialists,  and  as  prop- 
agandists we  address  this  prefatory  statement  to  particular 
groups  of  men  as  follows  : 

(a)  To  those  having  a  general  interest  in  education,  we  recommend  Bill's 
School  and  Mine  as  well  worth  reading,  and  re-reading. 

(6)  To  those  specially  concerned  with  scientific  and  technical  education, 
we  recommend  the  philosophy  and  the  biting  humor  that  is  scattered 
throughout  the  Calendar  of  Leading  Experiments. 

(c)  To  teachers  of  physics  and  to  teachers  of  electrical  engeneering.    The 

earlier  books  on  these  subjects  are  widely  known,  but  General  Physics 
(1916)  and  Elements  of  Electrical  Engineering  Vol.  I.  (1917)  represent 
wide  departures  from  the  earlier  books,  and  they  are,  we  believe,  great 
improvements  thereon  from  a  teaching  point  of  view. 

(d)  To  teachers  of  mathematics  we  earnestly  recommend  a  careful  examina- 

tion of  our  Elements  of  Calculus.  This  book  does  not  side-step  the  rigors 
of  mathematics.  It  represents  a  point  of  view  which  is  meeting  with 
increase  of  approval,  and  more  than  approval,  among  those  who  are 
broadly  and  vitally  concerned  with  the  mathematical  sciences. 

(e)  To  teachers  in  trade  schools  and  night  schools  we  recommend  Elementary 

Statics  (No.  8)  and  Elementary  Electricity  and  Magnetism  (No.  9). 

(/)  To  students  of  the  non-mathematical  sciences.  Franklin,  MacNutt 
and  Charles  take  pride  in  the  fact  that  what  they  stand  for  is  quoted  in 
William  James'  book  on  Pragmatism  as  an  extreme  example  of  pragmatic 
philosophy.  The  references  which  are  brought  together  on  page  8  of 
this  circular  constitute  a  very  simple  and  highly  pragmatic  treatise  on 
the  philosophy  of  the  mathematical  sciences. 

South  Bethlehem,  Pennsylvania 

8^"  Any  book  in  this  list  will  be  sent  post  paid  on  approval,  to  be 
returned  post  paid  if  not  satisfactory. 


No.  1 

GENERAL  PHYSICS 

(1916} 

BY 

WM.  S.  FRANKLIN  AND  BARRY  MAcNUTT. 

A   TREATISE   ON   NATURAL   PHILOSOPHY 

Published  and  for  sale  by  McGraw-Hill  Book  Company 

Price  $2.75  post  paid 
A  Key  to  Problems  is  supplied  with  teacher's  desk  copies  of  this  book 

This  textbook,  with  its  brevity  and  clearness,  its  directness 
of  treatment  and  its  freedom  from  tedious  repetition  of  the 
commonplaces  of  High  School  Physics,  was  intended  to  do  away 
with  the  usual  excessive  amount  of  classroom  coaching  which 
is  the  despair  of  every  physics  teacher  who  aims  to  develop  the 
power  of  analytical  thinking  and  who  tries  therefore  to  hold 
his  students  accountable  for  mathematical  ideas,  mathematical 
formulations  (not  mathematical  formulas,  please  note)  and 
numerical  calculations.  And  the  use  of  the  book  in  the  class- 
room seems  to  show  that  it  does  in  some  measure  accomplish 
what  was  intended.  The  students  do  read  it,  it  tends  to  focus 
their  attention  on  fundamentals,  it  helps  to  an  unprecedented 
extent  (in  the  authors'  experience)  to  form  precise  ideas,  and  it 
enables  the  student  to  devour  numerical  problems. 

For  sale  by 
FRANKLIN,  MacNUTT  AND   CHARLES 


No.  2 

A  CALENDAR  OF  LEADING 
EXPERIMENTS 

(1918) 

BY 

WM.  S.  FRANKLIN  AND  BARRY  MAcNUIT 

Published  and  for  sale  by 

FRANKLIN,   MacNUTT   AND   CHARLES 
South  Bethlehem,  Pa. 

Price  $2.50  post  paid 

This  book  has  to  do  primarily  with  lecture  demonstrations  in 
physics.  The  authors  believe  that  High  School  Physics  would 
be  greatly  helped  by  the  use  of  more  class-room  demonstrations, 
they  believe  that  such  demonstrations  can  be  provided  for  at  a 
very  small  expense,  and  they  consider  that  many  of  the  demon- 
strations described  in  this  book  would  be  usable  in  the  High 
School.  The  book  has  an  Appendix  on  A  Visitors'  Laboratory 
of  Physics,  and  certainly  our  State  Colleges  and  High  Schools 
could  do  a  great  work  by  developing  this  phase  of  the  museum 
idea. 

Secondarily  this  book  sets  forth  the  possibilities  of  an  extended 
course  in  elementary  dynamics,  including  the  dynamics  of  wave 
motion ;  and  from  beginning  to  end  the  book  is  rilled  with  semi- 
humorous  reflections  on  the  problems  of  the  teacher  of  physics. 


No.  3 

AN  ELEMENTARY  TREATISE 
ON  CALCULUS 


BY 

FRANKLIN,   MAcNUTT  AND   CHARLES 
Published  by  the  authors.    Price  $2.00  post  paid 

Nearly  everyone  who  has  examined  this  book  has  been  favor- 
ably impressed  by  its  mathematical  quality  and  by  its  simplicity 
and  clearness.  It  does  not  side-step  mathematical  precision  and 
rigor,  and  yet  the  textual  discussion  is  vividly  intelligible  so  that 
the  student  can  reasonably  be  expected  to  understand  the  subject 
without  the  usual  excessive  amount  of  class-room  coaching.  It 
contains  more  than  enough  carefully  graded  problem  and  exercise 
work  to  keep  the  average  student  busy  for  a  year. 

"It  will  teach  many  a  mathematician  many  a  useful  thing  if 
only  the  mathematicians  will  take  heed  unto  it."  Professor 
E.  B.  Wilson,  Massachusetts  Institute  of  Technology. 

"It  seems  to  me  that  you  have  succeeded  in  making  a  book 
that  will  cause  enthusiasm  rather  than  discouragement  in  the 
student."  A  professor  of  mathematics. 


No.  4 

BILL'S  SCHOOL  AND  MINE 

(mi) 

A  COLLECTION 
OF  ESSAYS  ON  EDUCATION 

BY  WM.  S.  FRANKLIN 

Published  and  for  sale  by 

FRANKLIN,   MacNUTT   AND   CHARLES 

South  Bethlehem,  Pa. 

Printed  on  India  paper  and  beautifully  bound. 
Price  $1.00  post  paid 

These  essays  are  so  compact  and  so  forcible  that  the  Inde- 
pendent called  the  book  "A  Package  of  Dynamite,"  and  the 
reviewer  in  The  Elementary  School  Journal  (University  of  Chi- 
cago) says  "Impossible  to  read  these  essays  lying  down." 

From  Nature  (London),  May  10,  1917.  "This  new  edition  of 
Professor  Franklin's  brightly  written  essays,  with  their  advocacy 
of  education  in  the  '  Land  of  Out-of-Doors '  and  of  the  claims  of 
sensible  science  to  a  prominent  place  in  school  curricula,  is  en- 
riched by  a  new  essay  on  Education  after  the  War." 

"These  brief  papers  are  written  in  an  entertaining  style  that 
is  gripping  and  interesting.  .  .  .  There  is  much  that  is  good  and 
much  that  is  of  value  in  a  careful  study  of  this  purposeful  little 
volume."  School  and  Science  Review. 

"Yesterday  I  read  every  word  of  your  book — how  splendid  it  is!    Two 
things  I  wish,  yea  three,  I  desire — 
First,  that  every  American  should  read  your  book. 
Second,  that  I  had  had  your  experiences. 

Third,  that  I  could  have  gone  to  school  to  you  in  Mathematics  and  Science, 
You  are  right  in  your  Science-teaching  ideas — absolutely  right!     But  we 
both  agree  that  teaching  is  great  fun,  what? 

Yours 

WM.  LYON  PHELPS  of  Yale  University 


COMPLETE  LIST 
OF  FRANKLIN  AND  MacNUTT  BOOKS 


1.  General  Physics.     McGraw-Hill  Book  Co.,  1916. Price  $2.75 

2.  Calendar    of    Leading    Experiments.     Franklin, 

MacNutt  and  Charles,  1918 "       2.50 

5.  Mechanic  sand  Heat.     TheMacmillanCo.,  1910,     "       1.75 

6.  Elements  of  Electricity  and  Magnetism.    The 

Macmillan  Co.,  1909 "        1.60 

7.  Light  and  Sound.    The  Macmillan  Co.,  1909.  .  .     "       1.60 

NOTE — Books  5,  6  and  7  constitute  a  fairly  complete  treatise  on 
the  elements  of  physics. 

8.  Elementary  Statics  (Mechanics).   Franklin,  Mac- 

Nutt and  Charles,  1915 "       0.50 

NOTE — This  book  is  in  pamphlet  form. 

9.  Elementary    Electricity    and    Magnetism.    The 

Macmillan  Co.,  1914 "        1.25 

NOTE — This  book  is  suitable  for  trade  schools  and  night  schools. 

10.  Advanced  Electricity  and  Magnetism.    The  Mac- 

millan Co.,  1915 "       2.00 

NOTE — Every  student  of  electrical  engineering  needs  something  be- 
yond the  bare  elements  of  electricity  and  magnetism,  and  this  book 
is  intended  to  supply  this  need.  Contents  areas  fol'ows:  Pages 
1-74  summary  of  elements,  pages  75-103  magnetism  of  iron;  pages 
104-120  ship's  magnetism  and  the  compensation  of  the  compass; 
pages  121-164  electrostatics;  pages  165-192  the  theory  of  poten- 
tial; pages  193-273  electric  waves;  and  pages  274-297  the  electron 
theory. 

Practical  Physics.  By  Franklin,  Crawford  and 
MacNutt.  TheMacmillanCo.,  1903.  A  Labora- 
tory Manual  of  Physics  in  three  volumes. 

11.  Volume    I.      Precise    Measurements.      Me- 

chanics and  Heat "        1.25 

12.  Volume    II.       Elementary    and    Advanced 

Measurements  in  Electricity  and  Magne- 
tism      "        1.25 

13.  Volume    III.     Photometry.     Measurements 

in  Light  and  Sound .90 

14.  Simple  Tables  for  students  of  physics  and  chemistry. 

Price  per  100 2.50 

For  Sale  by  the  Publishers  and  by 

FRANKLIN,  MacNUTT  AND  CHARLES 

South  Bethlehem,  Pa. 


Books  on  Electrical  Engineering 

For  sale  by 
FRANKLIN,   MacNUTT  AND   CHARLES 

and  also  by  the  publishers,  The  Macmillan  Co.,  N.  Y.  City 

ELEMENTS    OF   ELECTRICAL   ENGINEERING. 

Franklin  and  Esty. 

15.  Vol.  I.  Direct  Currents  1906 Price  $4.50 

16.  Vol.  II.  Alternating  Currents  1908 "       3.50 

ELEMENTS    OF   ELECTRICAL   ENGINEERING. 

W.  S.  Franklin. 

17.  Vol.  I.  D.    C.   and   A.    C.   Machines  and 
Systems  1917 "       4.50 

18.  Vol.  II.  Electric  Lighting  and  Miscellaneous 
Applications  1912 "       2.50 

Note.    A  Key  to  Problems  is  supplied  with  teachers'  desk 
copies  of  these  two  books. 

DYNAMO    LABORATORY    MANUAL.      Franklin 
and  Esty. 

19.  Vol.  I.  Direct  Currents  1906 "       1.75 

20.  Vol.  II.  In  preparation. 

21.  DYNAMOS    AND    MOTORS.     Franklin    and 

Esty,   1909 "       4.00 

This  volume  contains  the  portions  of  Franklin  and  Esty's 
Elements  which  relate  to  D.  C.  and  A.  C.  machines. 

22.  ELECTRIC   WAVES.     W.  S.  Franklin,  1909..     "       3.00 

The  more  elementary  portions  of  this  volume  are  given 
in  a  carefully  revised  form  on  pages  193-273  of  Frank- 
lin and  MacNutt's  Advanced  Electricity  and  Magnetism, 
book  No.  10  in  the  above  list. 


Many  biologists  and  many  purely  experimental  chemists  would  like  to 
have  a  clear  insight  into  the  methods  of  the  mathematical  sciences  ;  but  every 
science  is  now  presented  as  for  the  narrow  specialist,  or  for  the  dilettante,  and 
the  student  must  choose  everything  in  a  particular  field,  or  nothing.  An  exclu- 
sive interest  in  a  particular  science  seems  to  be  the  only  interest  that  is  recog- 
nized as  serious. 

The  following  references  bring  together  a  simple  and  highly 
pragmatic  treatise  on 

THE  PHILOSOPHY  OF  THE 

MATHEMATICAL  SCIENCES 

and  with  the  help  of  this  simple  treatise,  which  it  is  hoped  some 
time  to  publish  separately,  any  mature  student  can  get  a  clear 
understanding  of  the  methods  of  the  mathematical  sciences  with 
a  very  moderate  expenditure  of  time  and  effort. 

The  infinitesimal  calculus.  See  pages  593-597  of  book  No.  i  for  a 
discussion  of  the  meaning  of  the  infinitesimal  calculus,  and  see  chapter  one  of 
book  No.  3  (40  pages)  for  a  precise  and  clearly  intelligible  survey  of  differential 
and  integral  calculus.  This  chapter  is  presented  in  carefully  revised  form  in 
the  Supplement  which  accompanies  the  book. 

Methods  in  physics.  The  four  fundamental  methods  in  physics  are  (a) 
The  method  of  mechanics,  (b)  The  method  of  thermodynamics,  (c)  The  method 
of  atomics  (the  atomic  theory),  and  (d)  The  method  of  statistics. 

The  philosophy  of  elementary  mechanics  is  discussed  at  some  length  in 
book  No.  2.  See  pages  119-123  of  book  No.  i  for  a  discussion  of  the  atomic 
theory  in  contrast  with  thermodynamics;  see  pages  322-325  of  book  No.  I  for 
a  discussion  of  the  method  of  mechanics  as  contrasted  with  the  method  of 
atomics  and  see  a  very  simple  and  suggestive  discussion  of  the  use  of  the 
statistical  method  in  physics  in  Science,  pages  158-162,  August  4,  1916. 

The  conservation  of  energy  and  the  law  of  entropy.  Any  scientific 
work,  to  be  fruitful,  must  be  to  some  extent  conditioned  by  the  great  generali- 
zations of  physics;  but  these  generalizations  need  to  be  tempered,  in  their  wide 
acceptance,  by  a  full  appreciation  of  the  fact  that  they  are  very  largely  of 
postulate  content.  The  great  value  of  precise  ideas  (mathematical  ideas)  is 
that  they  open  the  mind  for  the  perception  of  the  simplest  evidences  of  a  sub- 
ject, and  mathematical  thinking  is  a  necessity  in  the  mathematical  sciences; 
but  any  kind  of  an  idea  unconditionally  held  closes  the  mind  completely  to 
contrary  evidences.  This  matter  is  discussed  somewhat  naively  on  pages  75- 
90  and  99-101  of  book  No.  4. 

The  law  of  entropy  is,  perhaps,  rigorously  true  for  a  world  in  which  no 
organizing  agencies  are  at  work  coordinating  things  which  are  apart  in  time  and 
apart  in  space,  but  living  forms  seem  to  be  subject  to  the  law  of  entropy  only  in 
that  the  easiest  line  of  delevopment  of  an  organism  which  must  use  free  energy 
is  the  line  of  development  in  which  an  excessive  burden  of  coordination  is  not 
necessary  at  the  beginning. 

•The  conservation  of  energy  is  discussed  on  pages  60-70  and  119-125  of 
book  No.  i. 

The  law  of  entropy  (the  second  law  of  thermadynamics)  is  discussed  on 
pages  153-168  of  book  No.  i. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


REC'D  LD 


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UU  i — &- 


4Dec'62GR 


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